Theorem scalar_sum_distribution 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 scalar_sum_distribution

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

Passed sanity check: scalar_sum_distribution matches stored_expr

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

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