Expression of type Implies

from the theory of proveit.linear_algebra.addition

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 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.addition import vec_summation_b1toj_fQ
from proveit.logic import Forall, Implies, InSet

# build up the expression from sub-expressions
expr = Implies(Forall(instance_param_or_params = [b_1_to_j], instance_expr = InSet(f__b_1_to_j, V), condition = Q__b_1_to_j), InSet(vec_summation_b1toj_fQ, V)).with_wrapping_at(2)

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())

\begin{array}{c} \begin{array}{l} \left[\forall_{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) \in V\right)\right] \Rightarrow  \\ \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 V\right) \end{array} \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 15 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	9, 18
8	Lambda	parameters: 23 body: 10
9	Operation	operator: 11 operand: 14
10	Conditional	value: 13 condition: 20
11	Literal
12	ExprTuple	14
13	Operation	operator: 15 operands: 16
14	Lambda	parameters: 23 body: 17
15	Literal
16	ExprTuple	19, 18
17	Conditional	value: 19 condition: 20
18	Variable
19	Operation	operator: 21 operands: 23
20	Operation	operator: 22 operands: 23
21	Variable
22	Variable
23	ExprTuple	24
24	ExprRange	lambda_map: 25 start_index: 26 end_index: 27
25	Lambda	parameter: 31 body: 28
26	Literal
27	Variable
28	IndexedVar	variable: 29 index: 31
29	Variable
30	ExprTuple	31
31	Variable