Theorem association 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 association

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

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