Theorem tensor_prod_distribution_over_add of type Forall

from the theory of proveit.linear_algebra.tensors

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.linear_algebra.tensors import tensor_prod_distribution_over_add

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

Passed sanity check: tensor_prod_distribution_over_add matches stored_expr

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

\forall_{K}~\left[\forall_{i, j, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)}~\\
\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}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} +  b_{2} +  \ldots +  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) \in V\right) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} \left(b_{1} +  b_{2} +  \ldots +  b_{j}\right){\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) =  \\ \left(\left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{1}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) +  \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{2}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right) +  \ldots +  \left(a_{1} {\otimes}  a_{2} {\otimes}  \ldots {\otimes}  a_{i} {\otimes} b_{j}{\otimes} c_{1} {\otimes}  c_{2} {\otimes}  \ldots {\otimes}  c_{k}\right)\right) \end{array} \end{array}\right) \end{array} \end{array}\right)\end{array}\right]\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	parameter: 37 body: 3
3	Operation	operator: 25 operand: 5
4	ExprTuple	5
5	Lambda	parameters: 6 body: 7
6	ExprTuple	65, 57, 70
7	Conditional	value: 8 condition: 9
8	Operation	operator: 25 operand: 13
9	Operation	operator: 11 operands: 12
10	ExprTuple	13
11	Literal
12	ExprTuple	14, 15, 16
13	Lambda	parameter: 44 body: 18
14	Operation	operator: 40 operands: 19
15	Operation	operator: 40 operands: 20
16	Operation	operator: 40 operands: 21
17	ExprTuple	44
18	Conditional	value: 22 condition: 23
19	ExprTuple	65, 24
20	ExprTuple	57, 24
21	ExprTuple	70, 24
22	Operation	operator: 25 operand: 29
23	Operation	operator: 27 operands: 28
24	Literal
25	Literal
26	ExprTuple	29
27	Literal
28	ExprTuple	44, 30
29	Lambda	parameters: 31 body: 32
30	Operation	operator: 33 operand: 37
31	ExprTuple	61, 54, 63
32	Operation	operator: 35 operands: 36
33	Literal
34	ExprTuple	37
35	Literal
36	ExprTuple	38, 39
37	Variable
38	Operation	operator: 40 operands: 41
39	Operation	operator: 42 operands: 43
40	Literal
41	ExprTuple	45, 44
42	Literal
43	ExprTuple	45, 46
44	Variable
45	Operation	operator: 58 operands: 47
46	Operation	operator: 51 operands: 48
47	ExprTuple	61, 49, 63
48	ExprTuple	50
49	Operation	operator: 51 operands: 52
50	ExprRange	lambda_map: 53 start_index: 69 end_index: 57
51	Literal
52	ExprTuple	54
53	Lambda	parameter: 72 body: 55
54	ExprRange	lambda_map: 56 start_index: 69 end_index: 57
55	Operation	operator: 58 operands: 59
56	Lambda	parameter: 77 body: 60
57	Variable
58	Literal
59	ExprTuple	61, 62, 63
60	IndexedVar	variable: 66 index: 77
61	ExprRange	lambda_map: 64 start_index: 69 end_index: 65
62	IndexedVar	variable: 66 index: 72
63	ExprRange	lambda_map: 68 start_index: 69 end_index: 70
64	Lambda	parameter: 77 body: 71
65	Variable
66	Variable
67	ExprTuple	72
68	Lambda	parameter: 77 body: 73
69	Literal
70	Variable
71	IndexedVar	variable: 74 index: 77
72	Variable
73	IndexedVar	variable: 75 index: 77
74	Variable
75	Variable
76	ExprTuple	77
77	Variable