Theorem distribution_over_vectors 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 distribution_over_vectors

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

Passed sanity check: distribution_over_vectors matches stored_expr

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

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