Theorem distribute_through_summation of type Forall

from the theory of proveit.numbers.multiplication

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.numbers.multiplication import distribute_through_summation

# check that the built expression is the same as the stored expression
assert distribute_through_summation.expr == stored_expr
assert distribute_through_summation.expr._style_id == stored_expr._style_id
print("Passed sanity check: distribute_through_summation matches stored_expr")

Passed sanity check: distribute_through_summation matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{i \in \mathbb{N}, j \in \mathbb{N}^+, k \in \mathbb{N}}~\left[\forall_{f, Q}~\left(\begin{array}{c} \begin{array}{l} \left[\forall_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(f\left(b_{1}, b_{2}, \ldots, b_{j}\right) \in \mathbb{C}\right)\right] \Rightarrow  \\ \left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right]\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right] \end{array} \end{array}\right)\end{array}\right] \end{array} \end{array}\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 25 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	70, 81, 74
4	Conditional	value: 5 condition: 6
5	Operation	operator: 25 operand: 9
6	Operation	operator: 38 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11, 12
9	Lambda	parameters: 13 body: 14
10	Operation	operator: 56 operands: 15
11	Operation	operator: 56 operands: 16
12	Operation	operator: 56 operands: 17
13	ExprTuple	71, 68
14	Operation	operator: 18 operands: 19
15	ExprTuple	70, 21
16	ExprTuple	81, 20
17	ExprTuple	74, 21
18	Literal
19	ExprTuple	22, 23
20	Literal
21	Literal
22	Operation	operator: 25 operand: 27
23	Operation	operator: 25 operand: 28
24	ExprTuple	27
25	Literal
26	ExprTuple	28
27	Lambda	parameters: 72 body: 29
28	Lambda	parameters: 30 body: 31
29	Conditional	value: 32 condition: 64
30	ExprTuple	65, 67
31	Conditional	value: 33 condition: 34
32	Operation	operator: 56 operands: 35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 39
35	ExprTuple	66, 60
36	Literal
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	42, 43
40	Operation	operator: 62 operands: 44
41	Operation	operator: 52 operand: 49
42	ExprRange	lambda_map: 46 start_index: 80 end_index: 70
43	ExprRange	lambda_map: 47 start_index: 80 end_index: 74
44	ExprTuple	65, 48, 67
45	ExprTuple	49
46	Lambda	parameter: 86 body: 50
47	Lambda	parameter: 86 body: 51
48	Operation	operator: 52 operand: 58
49	Lambda	parameters: 72 body: 54
50	Operation	operator: 56 operands: 55
51	Operation	operator: 56 operands: 57
52	Literal
53	ExprTuple	58
54	Conditional	value: 59 condition: 64
55	ExprTuple	75, 60
56	Literal
57	ExprTuple	77, 60
58	Lambda	parameters: 72 body: 61
59	Operation	operator: 62 operands: 63
60	Literal
61	Conditional	value: 66 condition: 64
62	Literal
63	ExprTuple	65, 66, 67
64	Operation	operator: 68 operands: 72
65	ExprRange	lambda_map: 69 start_index: 80 end_index: 70
66	Operation	operator: 71 operands: 72
67	ExprRange	lambda_map: 73 start_index: 80 end_index: 74
68	Variable
69	Lambda	parameter: 86 body: 75
70	Variable
71	Variable
72	ExprTuple	76
73	Lambda	parameter: 86 body: 77
74	Variable
75	IndexedVar	variable: 78 index: 86
76	ExprRange	lambda_map: 79 start_index: 80 end_index: 81
77	IndexedVar	variable: 82 index: 86
78	Variable
79	Lambda	parameter: 86 body: 83
80	Literal
81	Variable
82	Variable
83	IndexedVar	variable: 84 index: 86
84	Variable
85	ExprTuple	86
86	Variable