Show the Proof

import proveit
# Automation is not needed when only showing a stored proof:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%show_proof

	step type	requirements	statement
0	instantiation	1, 2, 3, 4	⊢
	: , : , : , :
1	reference	26	⊢
2	instantiation	213, 5, 6, 7*	⊢
	: , : , :
3	instantiation	226, 8	⊢
	: , : , :
4	instantiation	192	⊢
	:
5	instantiation	16, 24, 222, 273, 224, 42, 57, 21, 9	⊢
	: , : , : , : , : , : , : , : , : , :
6	instantiation	226, 10	⊢
	: , : , :
7	instantiation	11, 12, 13, 24, 25, 14*	⊢
	: , : , : , : , :
8	instantiation	99, 15	⊢
	: , :
9	instantiation	56, 57, 25, 22	⊢
	: , : , : , :
10	instantiation	16, 25, 273, 222, 224, 42, 57, 21, 22	⊢
	: , : , : , : , : , : , : , : , : , :
11	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.doubly_scaled_as_singly_scaled
12	instantiation	17, 282, 19, 42, 57	⊢
	: , : , :
13	instantiation	18, 282, 19, 42, 57, 20, 21, 22	⊢
	: , : , : , :
14	instantiation	23, 24, 25	⊢
	: , :
15	instantiation	26, 27, 28, 29	⊢
	: , : , : , :
16	theorem		⊢
	proveit.linear_algebra.tensors.factor_scalar_from_tensor_prod
17	theorem		⊢
	proveit.linear_algebra.tensors.tensor_prod_of_vec_spaces_is_vec_space
18	theorem		⊢
	proveit.linear_algebra.tensors.tensor_prod_is_in_tensor_prod_space
19	instantiation	236	⊢
	: , :
20	instantiation	236	⊢
	: , :
21	instantiation	30, 42, 31, 32	⊢
	: , : , : , :
22	modus ponens	33, 34	⊢
23	axiom		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
24	instantiation	97, 235, 35, 36	⊢
	: , :
25	instantiation	97, 235, 37, 38	⊢
	: , :
26	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
27	instantiation	213, 39, 40	⊢
	: , : , :
28	instantiation	192	⊢
	:
29	instantiation	99, 41	⊢
	: , :
30	theorem		⊢
	proveit.linear_algebra.addition.binary_closure
31	theorem		⊢
	proveit.physics.quantum.algebra.ket_zero_in_qubit_space
32	instantiation	56, 42, 43, 44	⊢
	: , : , : , :
33	instantiation	45, 280, 57	⊢
	: , : , : , : , : , :
34	generalization	46	⊢
35	instantiation	166, 253, 47	⊢
	: , :
36	instantiation	48, 105, 103	⊢
	: , : , :
37	instantiation	166, 253, 79	⊢
	: , :
38	instantiation	48, 109, 107	⊢
	: , : , :
39	instantiation	226, 49	⊢
	: , : , :
40	instantiation	201, 235, 50, 51, 52*	⊢
	: , :
41	instantiation	213, 53, 54	⊢
	: , : , :
42	instantiation	74, 282	⊢
	:
43	instantiation	166, 76, 55	⊢
	: , :
44	theorem		⊢
	proveit.physics.quantum.algebra.ket_one_in_qubit_space
45	theorem		⊢
	proveit.linear_algebra.addition.summation_closure
46	instantiation	56, 57, 58, 59	,  ⊢
	: , : , : , :
47	instantiation	233, 60, 61	⊢
	: , :
48	theorem		⊢
	proveit.logic.equality.sub_left_side_into
49	instantiation	213, 62, 63	⊢
	: , : , :
50	instantiation	168, 104, 108	⊢
	: , :
51	instantiation	64, 285, 65, 66, 67	⊢
	: , :
52	instantiation	213, 68, 69	⊢
	: , : , :
53	instantiation	226, 70	⊢
	: , : , :
54	instantiation	226, 71	⊢
	: , : , :
55	instantiation	132, 72, 73	⊢
	: , : , :
56	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
57	instantiation	74, 75	⊢
	:
58	instantiation	166, 76, 77	,  ⊢
	: , :
59	instantiation	78, 257, 209	,  ⊢
	: , :
60	instantiation	97, 220, 253, 202	⊢
	: , :
61	instantiation	205, 79	⊢
	:
62	instantiation	226, 173	⊢
	: , : , :
63	instantiation	213, 80, 81	⊢
	: , : , :
64	theorem		⊢
	proveit.numbers.multiplication.mult_not_eq_zero
65	instantiation	236	⊢
	: , :
66	instantiation	82, 104, 105	⊢
	:
67	instantiation	82, 108, 109	⊢
	:
68	instantiation	226, 83	⊢
	: , : , :
69	instantiation	213, 84, 85	⊢
	: , : , :
70	instantiation	213, 86, 87	⊢
	: , : , :
71	instantiation	213, 88, 89	⊢
	: , : , :
72	instantiation	168, 135, 90	⊢
	: , :
73	instantiation	213, 91, 92	⊢
	: , : , :
74	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
75	instantiation	93, 285, 245	⊢
	: , :
76	instantiation	283, 262, 94	⊢
	: , : , :
77	instantiation	132, 95, 96	,  ⊢
	: , : , :
78	theorem		⊢
	proveit.physics.quantum.algebra.num_ket_in_register_space
79	instantiation	97, 234, 253, 202	⊢
	: , :
80	instantiation	226, 98	⊢
	: , : , :
81	instantiation	99, 100	⊢
	: , :
82	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
83	instantiation	101, 104, 108, 164, 105, 109, 144*, 147*	⊢
	: , : , :
84	instantiation	194, 273, 285, 222, 102, 224, 235, 145, 149	⊢
	: , : , : , : , : , :
85	instantiation	206, 222, 285, 224, 102, 145, 149	⊢
	: , : , : , :
86	instantiation	226, 103	⊢
	: , : , :
87	instantiation	201, 235, 104, 105, 106*	⊢
	: , :
88	instantiation	226, 107	⊢
	: , : , :
89	instantiation	201, 235, 108, 109, 110*	⊢
	: , :
90	instantiation	132, 111, 112	⊢
	: , : , :
91	instantiation	194, 273, 136, 222, 113, 224, 135, 169, 170, 131	⊢
	: , : , : , : , : , :
92	instantiation	194, 222, 285, 136, 224, 137, 113, 253, 154, 169, 170, 131	⊢
	: , : , : , : , : , :
93	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
94	instantiation	283, 187, 114	⊢
	: , : , :
95	instantiation	168, 135, 115	,  ⊢
	: , :
96	instantiation	213, 116, 117	,  ⊢
	: , : , :
97	theorem		⊢
	proveit.numbers.division.div_complex_closure
98	instantiation	118, 191, 238	⊢
	: , :
99	theorem		⊢
	proveit.logic.equality.equals_reversal
100	instantiation	119, 253, 120, 121	⊢
	: , : , :
101	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
102	instantiation	236	⊢
	: , :
103	instantiation	226, 122	⊢
	: , : , :
104	instantiation	123, 253	⊢
	:
105	instantiation	127, 261, 124	⊢
	: , :
106	instantiation	213, 125, 126	⊢
	: , : , :
107	instantiation	226, 190	⊢
	: , : , :
108	instantiation	166, 253, 191	⊢
	: , :
109	instantiation	127, 261, 128	⊢
	: , :
110	instantiation	213, 129, 130	⊢
	: , : , :
111	instantiation	168, 151, 131	⊢
	: , :
112	instantiation	194, 222, 285, 273, 224, 152, 169, 170, 131	⊢
	: , : , : , : , : , :
113	instantiation	155	⊢
	: , : , :
114	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
115	instantiation	132, 133, 134	,  ⊢
	: , : , :
116	instantiation	194, 273, 136, 222, 138, 224, 135, 169, 170, 153	,  ⊢
	: , : , : , : , : , :
117	instantiation	194, 222, 285, 136, 224, 137, 138, 253, 154, 169, 170, 153	,  ⊢
	: , : , : , : , : , :
118	theorem		⊢
	proveit.numbers.addition.commutation
119	theorem		⊢
	proveit.numbers.exponentiation.product_of_pos_powers
120	instantiation	283, 139, 268	⊢
	: , : , :
121	instantiation	283, 139, 229	⊢
	: , : , :
122	instantiation	213, 140, 141	⊢
	: , : , :
123	theorem		⊢
	proveit.numbers.exponentiation.sqrt_complex_closure
124	instantiation	142, 143, 261	⊢
	: , :
125	instantiation	226, 144	⊢
	: , : , :
126	instantiation	148, 145	⊢
	:
127	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
128	instantiation	283, 146, 229	⊢
	: , : , :
129	instantiation	226, 147	⊢
	: , : , :
130	instantiation	148, 149	⊢
	:
131	instantiation	283, 262, 150	⊢
	: , : , :
132	theorem		⊢
	proveit.logic.equality.sub_right_side_into
133	instantiation	168, 151, 153	,  ⊢
	: , :
134	instantiation	194, 222, 285, 273, 224, 152, 169, 170, 153	,  ⊢
	: , : , : , : , : , :
135	instantiation	168, 253, 154	⊢
	: , :
136	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
137	instantiation	236	⊢
	: , :
138	instantiation	155	⊢
	: , : , :
139	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
140	instantiation	213, 156, 157	⊢
	: , : , :
141	instantiation	213, 158, 159	⊢
	: , : , :
142	theorem		⊢
	proveit.numbers.division.div_rational_nonzero_closure
143	instantiation	283, 269, 160	⊢
	: , : , :
144	instantiation	163, 253, 249, 164, 202, 161*	⊢
	: , : , :
145	instantiation	166, 253, 162	⊢
	: , :
146	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
147	instantiation	163, 253, 204, 164, 202, 165*	⊢
	: , : , :
148	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
149	instantiation	166, 253, 177	⊢
	: , :
150	instantiation	283, 271, 167	⊢
	: , : , :
151	instantiation	168, 169, 170	⊢
	: , :
152	instantiation	236	⊢
	: , :
153	instantiation	283, 262, 171	,  ⊢
	: , : , :
154	instantiation	283, 262, 172	⊢
	: , : , :
155	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
156	instantiation	226, 173	⊢
	: , : , :
157	instantiation	226, 174	⊢
	: , : , :
158	instantiation	175, 222, 285, 273, 224, 176, 191, 238, 177	⊢
	: , : , : , : , : , :
159	instantiation	178, 191, 238, 179	⊢
	: , : , :
160	instantiation	283, 277, 280	⊢
	: , : , :
161	instantiation	180, 238, 235, 225*	⊢
	: , :
162	instantiation	283, 262, 181	⊢
	: , : , :
163	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_real_power
164	instantiation	283, 271, 182	⊢
	: , : , :
165	instantiation	213, 183, 184	⊢
	: , : , :
166	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
167	instantiation	283, 278, 231	⊢
	: , : , :
168	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
169	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
170	instantiation	283, 262, 185	⊢
	: , : , :
171	instantiation	283, 271, 186	,  ⊢
	: , : , :
172	instantiation	283, 187, 188	⊢
	: , : , :
173	instantiation	201, 220, 253, 202, 189*	⊢
	: , :
174	instantiation	226, 190	⊢
	: , : , :
175	theorem		⊢
	proveit.numbers.addition.disassociation
176	instantiation	236	⊢
	: , :
177	instantiation	205, 191	⊢
	:
178	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
179	instantiation	192	⊢
	:
180	theorem		⊢
	proveit.numbers.negation.pos_times_neg
181	instantiation	193, 249	⊢
	:
182	instantiation	283, 278, 232	⊢
	: , : , :
183	instantiation	194, 222, 285, 273, 224, 207, 238, 234, 195	⊢
	: , : , : , : , : , :
184	instantiation	196, 285, 222, 207, 224, 238, 234, 235, 197*	⊢
	: , : , : , : , :
185	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
186	instantiation	283, 278, 198	,  ⊢
	: , : , :
187	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
188	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
189	instantiation	213, 199, 200	⊢
	: , : , :
190	instantiation	201, 234, 253, 202, 203*	⊢
	: , :
191	instantiation	283, 262, 204	⊢
	: , : , :
192	axiom		⊢
	proveit.logic.equality.equals_reflexivity
193	theorem		⊢
	proveit.numbers.negation.real_closure
194	theorem		⊢
	proveit.numbers.multiplication.disassociation
195	instantiation	205, 235	⊢
	:
196	theorem		⊢
	proveit.numbers.multiplication.mult_neg_any
197	instantiation	206, 285, 222, 207, 224, 238, 234	⊢
	: , : , : , :
198	instantiation	283, 208, 209	,  ⊢
	: , : , :
199	instantiation	226, 227	⊢
	: , : , :
200	instantiation	213, 210, 211	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.division.div_as_mult
202	instantiation	212, 282	⊢
	:
203	instantiation	213, 214, 215	⊢
	: , : , :
204	instantiation	283, 271, 216	⊢
	: , : , :
205	theorem		⊢
	proveit.numbers.negation.complex_closure
206	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
207	instantiation	236	⊢
	: , :
208	instantiation	217, 218, 219	⊢
	: , :
209	assumption		⊢
210	instantiation	228, 220, 238	⊢
	: , :
211	instantiation	221, 273, 285, 222, 223, 224, 238, 234, 235, 225*	⊢
	: , : , : , : , : , :
212	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
213	axiom		⊢
	proveit.logic.equality.equals_transitivity
214	instantiation	226, 227	⊢
	: , : , :
215	instantiation	228, 234, 238	⊢
	: , :
216	instantiation	283, 267, 229	⊢
	: , : , :
217	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
218	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
219	instantiation	230, 231, 232	⊢
	: , :
220	instantiation	233, 234, 235	⊢
	: , :
221	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
222	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
223	instantiation	236	⊢
	: , :
224	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
225	instantiation	237, 238	⊢
	:
226	axiom		⊢
	proveit.logic.equality.substitution
227	instantiation	239, 240, 280, 241*	⊢
	: , :
228	theorem		⊢
	proveit.numbers.multiplication.commutation
229	instantiation	242, 268, 243	⊢
	: , :
230	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
231	instantiation	244, 279, 245	⊢
	: , :
232	instantiation	246, 266	⊢
	:
233	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
234	instantiation	283, 262, 247	⊢
	: , : , :
235	instantiation	283, 262, 248	⊢
	: , : , :
236	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
237	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
238	instantiation	283, 262, 249	⊢
	: , : , :
239	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
240	instantiation	283, 250, 251	⊢
	: , : , :
241	instantiation	252, 253	⊢
	:
242	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
243	instantiation	283, 281, 257	⊢
	: , : , :
244	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
245	instantiation	283, 254, 257	⊢
	: , : , :
246	theorem		⊢
	proveit.numbers.negation.int_closure
247	instantiation	255, 256, 257	⊢
	: , : , :
248	instantiation	283, 271, 258	⊢
	: , : , :
249	instantiation	283, 271, 259	⊢
	: , : , :
250	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
251	instantiation	283, 260, 261	⊢
	: , : , :
252	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
253	instantiation	283, 262, 263	⊢
	: , : , :
254	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
255	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
256	instantiation	264, 265	⊢
	: , :
257	assumption		⊢
258	instantiation	283, 278, 266	⊢
	: , : , :
259	instantiation	283, 267, 268	⊢
	: , : , :
260	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
261	instantiation	283, 269, 270	⊢
	: , : , :
262	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
263	instantiation	283, 271, 272	⊢
	: , : , :
264	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
265	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
266	instantiation	283, 284, 273	⊢
	: , : , :
267	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
268	instantiation	274, 275, 276	⊢
	: , :
269	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
270	instantiation	283, 277, 282	⊢
	: , : , :
271	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
272	instantiation	283, 278, 279	⊢
	: , : , :
273	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
274	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
275	instantiation	283, 281, 280	⊢
	: , : , :
276	instantiation	283, 281, 282	⊢
	: , : , :
277	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
278	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
279	instantiation	283, 284, 285	⊢
	: , : , :
280	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
281	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
282	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
283	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
284	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
285	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements