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	⊢
	: , : , :
1	reference	118	⊢
2	instantiation	118, 4, 5	⊢
	: , : , :
3	instantiation	6, 26, 55, 7, 24, 8*	⊢
	: , : , :
4	instantiation	118, 9, 10	⊢
	: , : , :
5	instantiation	11, 187, 182, 55, 23, 24	⊢
	: , : , : , : , :
6	theorem		⊢
	proveit.physics.quantum.algebra.scalar_mult_factorization
7	instantiation	114	⊢
	: , : , :
8	instantiation	12, 55, 13	⊢
	: , :
9	instantiation	14, 55, 187, 109, 110, 15	⊢
	: , : , : , : , : , :
10	instantiation	22, 182, 176, 109, 42, 23, 110, 16	⊢
	: , : , : , : , : , :
11	theorem		⊢
	proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front
12	axiom		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
13	instantiation	78, 17, 29	⊢
	: , : , :
14	theorem		⊢
	proveit.physics.quantum.algebra.scalar_mult_absorption
15	instantiation	78, 24, 18	⊢
	: , : , :
16	instantiation	78, 19, 20	⊢
	: , : , :
17	instantiation	21, 67, 68, 32, 33	⊢
	: , : , : , :
18	instantiation	22, 187, 176, 109, 23, 110, 24	⊢
	: , : , : , : , : , :
19	instantiation	31, 67, 68, 25, 33	⊢
	: , : , : , :
20	instantiation	41, 26, 27	⊢
	: , : , :
21	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_ket_is_ket
22	theorem		⊢
	proveit.physics.quantum.algebra.qmult_disassociation
23	instantiation	124	⊢
	: , :
24	instantiation	78, 28, 29	⊢
	: , : , :
25	instantiation	66, 67, 68, 30	⊢
	: , : , :
26	theorem		⊢
	proveit.numbers.numerals.decimals.posnat3
27	instantiation	114	⊢
	: , : , :
28	instantiation	31, 67, 68, 32, 33	⊢
	: , : , : , :
29	instantiation	41, 176, 34	⊢
	: , : , :
30	instantiation	78, 35, 36	⊢
	: , : , :
31	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain
32	instantiation	66, 67, 68, 37	⊢
	: , : , :
33	modus ponens	38, 39	⊢
34	instantiation	124	⊢
	: , :
35	instantiation	43, 67, 68, 40, 44	⊢
	: , : , : , : , :
36	instantiation	41, 176, 42	⊢
	: , : , :
37	instantiation	43, 67, 68, 56, 44	⊢
	: , : , : , : , :
38	instantiation	45, 174, 51	⊢
	: , : , : , : , : , :
39	generalization	46	⊢
40	instantiation	66, 67, 68, 47	⊢
	: , : , :
41	axiom		⊢
	proveit.physics.quantum.algebra.multi_qmult_def
42	instantiation	124	⊢
	: , :
43	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_op_is_op
44	instantiation	48, 82, 49	⊢
	: , : , :
45	theorem		⊢
	proveit.linear_algebra.addition.summation_closure
46	instantiation	50, 51, 52, 53	⊢
	: , : , : , :
47	instantiation	54, 55, 67, 68, 56	⊢
	: , : , : , :
48	theorem		⊢
	proveit.physics.quantum.algebra.qmult_matrix_is_linmap
49	instantiation	146, 57, 58	⊢
	: , : , :
50	theorem		⊢
	proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
51	instantiation	59, 82	⊢
	:
52	instantiation	76, 60, 61	⊢
	: , :
53	instantiation	62, 184, 150	⊢
	: , :
54	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_complex_closure
55	instantiation	90, 63, 64, 65	⊢
	: , :
56	instantiation	66, 67, 68, 69	⊢
	: , : , :
57	instantiation	70, 82	⊢
	:
58	instantiation	71, 184	⊢
	:
59	theorem		⊢
	proveit.linear_algebra.complex_vec_set_is_vec_space
60	instantiation	185, 153, 72	⊢
	: , : , :
61	instantiation	95, 73, 74	⊢
	: , : , :
62	theorem		⊢
	proveit.physics.quantum.algebra.num_ket_in_register_space
63	instantiation	185, 153, 75	⊢
	: , : , :
64	instantiation	76, 145, 77	⊢
	: , :
65	instantiation	78, 79, 80	⊢
	: , : , :
66	theorem		⊢
	proveit.physics.quantum.algebra.qmult_op_is_linmap
67	instantiation	81, 82	⊢
	:
68	theorem		⊢
	proveit.linear_algebra.inner_products.complex_set_is_hilbert_space
69	instantiation	83, 184, 84	⊢
	: , :
70	theorem		⊢
	proveit.linear_algebra.matrices.unitaries_are_matrices
71	theorem		⊢
	proveit.physics.quantum.QFT.invFT_is_unitary
72	instantiation	185, 134, 85	⊢
	: , : , :
73	instantiation	121, 98, 86	⊢
	: , :
74	instantiation	118, 87, 88	⊢
	: , : , :
75	instantiation	185, 164, 89	⊢
	: , : , :
76	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
77	instantiation	90, 130, 145, 105	⊢
	: , :
78	theorem		⊢
	proveit.logic.equality.sub_left_side_into
79	instantiation	91, 152, 92	⊢
	: , :
80	instantiation	127, 93	⊢
	: , : , :
81	theorem		⊢
	proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space
82	instantiation	94, 182, 179	⊢
	: , :
83	theorem		⊢
	proveit.physics.quantum.algebra.num_bra_is_lin_map
84	assumption		⊢
85	theorem		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
86	instantiation	95, 96, 97	⊢
	: , : , :
87	instantiation	108, 187, 99, 109, 101, 110, 98, 122, 123, 112	⊢
	: , : , : , : , : , :
88	instantiation	108, 109, 182, 99, 110, 100, 101, 145, 113, 122, 123, 112	⊢
	: , : , : , : , : , :
89	instantiation	185, 173, 181	⊢
	: , : , :
90	theorem		⊢
	proveit.numbers.division.div_complex_closure
91	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_non_zero__not_zero
92	instantiation	185, 102, 103	⊢
	: , : , :
93	instantiation	104, 130, 145, 105, 106*	⊢
	: , :
94	theorem		⊢
	proveit.numbers.exponentiation.exp_natpos_closure
95	theorem		⊢
	proveit.logic.equality.sub_right_side_into
96	instantiation	121, 107, 112	⊢
	: , :
97	instantiation	108, 109, 182, 187, 110, 111, 122, 123, 112	⊢
	: , : , : , : , : , :
98	instantiation	121, 145, 113	⊢
	: , :
99	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
100	instantiation	124	⊢
	: , :
101	instantiation	114	⊢
	: , : , :
102	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
103	instantiation	115, 158, 116	⊢
	: , :
104	theorem		⊢
	proveit.numbers.division.div_as_mult
105	instantiation	117, 176	⊢
	:
106	instantiation	118, 119, 120	⊢
	: , : , :
107	instantiation	121, 122, 123	⊢
	: , :
108	theorem		⊢
	proveit.numbers.multiplication.disassociation
109	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
110	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
111	instantiation	124	⊢
	: , :
112	instantiation	185, 153, 125	⊢
	: , : , :
113	instantiation	185, 153, 126	⊢
	: , : , :
114	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
115	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
116	instantiation	185, 175, 184	⊢
	: , : , :
117	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
118	axiom		⊢
	proveit.logic.equality.equals_transitivity
119	instantiation	127, 128	⊢
	: , : , :
120	instantiation	129, 130, 131	⊢
	: , :
121	theorem		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
122	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
123	instantiation	185, 153, 132	⊢
	: , : , :
124	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
125	instantiation	185, 164, 133	⊢
	: , : , :
126	instantiation	185, 134, 135	⊢
	: , : , :
127	axiom		⊢
	proveit.logic.equality.substitution
128	instantiation	136, 137, 174, 138*	⊢
	: , :
129	theorem		⊢
	proveit.numbers.multiplication.commutation
130	instantiation	185, 153, 139	⊢
	: , : , :
131	instantiation	185, 153, 140	⊢
	: , : , :
132	theorem		⊢
	proveit.physics.quantum.QPE._phase_is_real
133	instantiation	185, 173, 141	⊢
	: , : , :
134	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
135	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
136	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
137	instantiation	185, 142, 143	⊢
	: , : , :
138	instantiation	144, 145	⊢
	:
139	instantiation	146, 147, 184	⊢
	: , : , :
140	instantiation	185, 164, 148	⊢
	: , : , :
141	instantiation	185, 149, 150	⊢
	: , : , :
142	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
143	instantiation	185, 151, 152	⊢
	: , : , :
144	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
145	instantiation	185, 153, 154	⊢
	: , : , :
146	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
147	instantiation	155, 156	⊢
	: , :
148	instantiation	185, 157, 158	⊢
	: , : , :
149	instantiation	159, 160, 161	⊢
	: , :
150	assumption		⊢
151	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
152	instantiation	185, 162, 163	⊢
	: , : , :
153	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
154	instantiation	185, 164, 165	⊢
	: , : , :
155	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
156	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
157	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
158	instantiation	166, 167, 168	⊢
	: , :
159	theorem		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
160	theorem		⊢
	proveit.numbers.number_sets.integers.zero_is_int
161	instantiation	169, 170, 171	⊢
	: , :
162	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
163	instantiation	185, 172, 176	⊢
	: , : , :
164	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
165	instantiation	185, 173, 178	⊢
	: , : , :
166	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
167	instantiation	185, 175, 174	⊢
	: , : , :
168	instantiation	185, 175, 176	⊢
	: , : , :
169	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
170	instantiation	177, 178, 179	⊢
	: , :
171	instantiation	180, 181	⊢
	:
172	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
173	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
174	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
175	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
176	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
177	theorem		⊢
	proveit.numbers.exponentiation.exp_int_closure
178	instantiation	185, 186, 182	⊢
	: , : , :
179	instantiation	185, 183, 184	⊢
	: , : , :
180	theorem		⊢
	proveit.numbers.negation.int_closure
181	instantiation	185, 186, 187	⊢
	: , : , :
182	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
183	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
184	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
185	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
186	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
187	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements