Theorem factorization_from_vec_sum_with_scalar_mult of type Forall

from the theory of proveit.linear_algebra.scalar_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.linear_algebra.scalar_multiplication import factorization_from_vec_sum_with_scalar_mult

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

Passed sanity check: factorization_from_vec_sum_with_scalar_mult matches stored_expr

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

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