Expression of type Conditional

from the theory of proveit.linear_algebra.scalar_multiplication

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import Conditional, K, V, v
from proveit.core_expr_types import Q__b_1_to_j, b_1_to_j, f__b_1_to_j
from proveit.linear_algebra import ScalarMult, VecSpaces, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Forall, Implies, InClass, InSet
from proveit.numbers import Sum

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
expr = Conditional(Forall(instance_param_or_params = [v], instance_expr = Implies(InSet(vec_summation_b1toj_fQ, K), Equals(ScalarMult(Sum(index_or_indices = sub_expr1, summand = f__b_1_to_j, condition = Q__b_1_to_j), v), VecSum(index_or_indices = sub_expr1, summand = ScalarMult(f__b_1_to_j, v), condition = Q__b_1_to_j)).with_wrapping_at(1)).with_wrapping_at(2), domain = V), InClass(V, VecSpaces(K)))

expr:

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

Passed sanity check: expr matches stored_expr

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

\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) \textrm{ if } V \underset{{\scriptscriptstyle c}}{\in} \textrm{VecSpaces}\left(K\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 7
2	Operation	operator: 5 operands: 6
3	Literal
4	ExprTuple	7
5	Literal
6	ExprTuple	20, 8
7	Lambda	parameter: 44 body: 10
8	Operation	operator: 11 operand: 26
9	ExprTuple	44
10	Conditional	value: 13 condition: 14
11	Literal
12	ExprTuple	26
13	Operation	operator: 15 operands: 16
14	Operation	operator: 21 operands: 17
15	Literal
16	ExprTuple	18, 19
17	ExprTuple	44, 20
18	Operation	operator: 21 operands: 22
19	Operation	operator: 23 operands: 24
20	Variable
21	Literal
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 30 operand: 37
26	Variable
27	Operation	operator: 40 operands: 29
28	Operation	operator: 30 operand: 33
29	ExprTuple	32, 44
30	Literal
31	ExprTuple	33
32	Operation	operator: 34 operand: 37
33	Lambda	parameters: 47 body: 36
34	Literal
35	ExprTuple	37
36	Conditional	value: 38 condition: 42
37	Lambda	parameters: 47 body: 39
38	Operation	operator: 40 operands: 41
39	Conditional	value: 43 condition: 42
40	Literal
41	ExprTuple	43, 44
42	Operation	operator: 45 operands: 47
43	Operation	operator: 46 operands: 47
44	Variable
45	Variable
46	Variable
47	ExprTuple	48
48	ExprRange	lambda_map: 49 start_index: 50 end_index: 51
49	Lambda	parameter: 55 body: 52
50	Literal
51	Variable
52	IndexedVar	variable: 53 index: 55
53	Variable
54	ExprTuple	55
55	Variable