Expression of type Forall

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 K, V, j, 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, InSet
from proveit.numbers import NaturalPos, Sum

# build up the expression from sub-expressions
sub_expr1 = [b_1_to_j]
expr = Forall(instance_param_or_params = [j], instance_expr = Forall(instance_param_or_params = [V], instance_expr = 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), domain = VecSpaces(K)), domain = NaturalPos)

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())

\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]

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: 15 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 63 body: 4
3	ExprTuple	63
4	Conditional	value: 5 condition: 6
5	Operation	operator: 15 operand: 9
6	Operation	operator: 33 operands: 8
7	ExprTuple	9
8	ExprTuple	63, 10
9	Lambda	parameter: 32 body: 12
10	Literal
11	ExprTuple	32
12	Conditional	value: 13 condition: 14
13	Operation	operator: 15 operand: 19
14	Operation	operator: 17 operands: 18
15	Literal
16	ExprTuple	19
17	Literal
18	ExprTuple	32, 20
19	Lambda	parameter: 56 body: 22
20	Operation	operator: 23 operand: 38
21	ExprTuple	56
22	Conditional	value: 25 condition: 26
23	Literal
24	ExprTuple	38
25	Operation	operator: 27 operands: 28
26	Operation	operator: 33 operands: 29
27	Literal
28	ExprTuple	30, 31
29	ExprTuple	56, 32
30	Operation	operator: 33 operands: 34
31	Operation	operator: 35 operands: 36
32	Variable
33	Literal
34	ExprTuple	37, 38
35	Literal
36	ExprTuple	39, 40
37	Operation	operator: 42 operand: 49
38	Variable
39	Operation	operator: 52 operands: 41
40	Operation	operator: 42 operand: 45
41	ExprTuple	44, 56
42	Literal
43	ExprTuple	45
44	Operation	operator: 46 operand: 49
45	Lambda	parameters: 59 body: 48
46	Literal
47	ExprTuple	49
48	Conditional	value: 50 condition: 54
49	Lambda	parameters: 59 body: 51
50	Operation	operator: 52 operands: 53
51	Conditional	value: 55 condition: 54
52	Literal
53	ExprTuple	55, 56
54	Operation	operator: 57 operands: 59
55	Operation	operator: 58 operands: 59
56	Variable
57	Variable
58	Variable
59	ExprTuple	60
60	ExprRange	lambda_map: 61 start_index: 62 end_index: 63
61	Lambda	parameter: 67 body: 64
62	Literal
63	Variable
64	IndexedVar	variable: 65 index: 67
65	Variable
66	ExprTuple	67
67	Variable