Theorem distribute_through_sum 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_sum

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

Passed sanity check: distribute_through_sum matches stored_expr

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

\forall_{i, j, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}, 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(b_{1} +  b_{2} +  \ldots +  b_{j}\right)\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{1}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{2}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \ldots +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{j}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right) \end{array} \end{array}\right)\end{array}\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	60, 52, 65
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 24 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	11, 12, 13
10	Lambda	parameters: 14 body: 15
11	Operation	operator: 46 operands: 16
12	Operation	operator: 46 operands: 17
13	Operation	operator: 46 operands: 18
14	ExprTuple	56, 48, 58
15	Conditional	value: 19 condition: 20
16	ExprTuple	60, 21
17	ExprTuple	52, 21
18	ExprTuple	65, 21
19	Operation	operator: 22 operands: 23
20	Operation	operator: 24 operands: 25
21	Literal
22	Literal
23	ExprTuple	26, 27
24	Literal
25	ExprTuple	28, 29, 30
26	Operation	operator: 53 operands: 31
27	Operation	operator: 41 operands: 32
28	ExprRange	lambda_map: 33 start_index: 64 end_index: 60
29	ExprRange	lambda_map: 34 start_index: 64 end_index: 52
30	ExprRange	lambda_map: 35 start_index: 64 end_index: 65
31	ExprTuple	56, 36, 58
32	ExprTuple	37
33	Lambda	parameter: 72 body: 38
34	Lambda	parameter: 72 body: 39
35	Lambda	parameter: 72 body: 40
36	Operation	operator: 41 operands: 42
37	ExprRange	lambda_map: 43 start_index: 64 end_index: 52
38	Operation	operator: 46 operands: 44
39	Operation	operator: 46 operands: 45
40	Operation	operator: 46 operands: 47
41	Literal
42	ExprTuple	48
43	Lambda	parameter: 67 body: 49
44	ExprTuple	66, 50
45	ExprTuple	55, 50
46	Literal
47	ExprTuple	68, 50
48	ExprRange	lambda_map: 51 start_index: 64 end_index: 52
49	Operation	operator: 53 operands: 54
50	Literal
51	Lambda	parameter: 72 body: 55
52	Variable
53	Literal
54	ExprTuple	56, 57, 58
55	IndexedVar	variable: 61 index: 72
56	ExprRange	lambda_map: 59 start_index: 64 end_index: 60
57	IndexedVar	variable: 61 index: 67
58	ExprRange	lambda_map: 63 start_index: 64 end_index: 65
59	Lambda	parameter: 72 body: 66
60	Variable
61	Variable
62	ExprTuple	67
63	Lambda	parameter: 72 body: 68
64	Literal
65	Variable
66	IndexedVar	variable: 69 index: 72
67	Variable
68	IndexedVar	variable: 70 index: 72
69	Variable
70	Variable
71	ExprTuple	72
72	Variable