Expression of type Implies

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 V, k
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, VecSum
from proveit.linear_algebra.addition import vec_summation_b1toj_fQ
from proveit.logic import Equals, Implies, InSet

# build up the expression from sub-expressions
expr = Implies(InSet(vec_summation_b1toj_fQ, V), Equals(VecSum(index_or_indices = [b_1_to_j], summand = ScalarMult(k, f__b_1_to_j), condition = Q__b_1_to_j), ScalarMult(k, vec_summation_b1toj_fQ)).with_wrapping_at(1)).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(\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) \Rightarrow  \\ \left(\begin{array}{c} \begin{array}{l} \left[\sum_{b_{1}, b_{2}, \ldots, b_{j}~|~Q\left(b_{1}, b_{2}, \ldots, b_{j}\right)}~\left(k \cdot f\left(b_{1}, b_{2}, \ldots, b_{j}\right)\right)\right] \\  = \left(k \cdot \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]\right) \end{array} \end{array}\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 operands: 6
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	15, 9
7	Literal
8	ExprTuple	10, 11
9	Variable
10	Operation	operator: 17 operand: 14
11	Operation	operator: 21 operands: 13
12	ExprTuple	14
13	ExprTuple	24, 15
14	Lambda	parameters: 29 body: 16
15	Operation	operator: 17 operand: 20
16	Conditional	value: 19 condition: 26
17	Literal
18	ExprTuple	20
19	Operation	operator: 21 operands: 22
20	Lambda	parameters: 29 body: 23
21	Literal
22	ExprTuple	24, 25
23	Conditional	value: 25 condition: 26
24	Variable
25	Operation	operator: 27 operands: 29
26	Operation	operator: 28 operands: 29
27	Variable
28	Variable
29	ExprTuple	30
30	ExprRange	lambda_map: 31 start_index: 32 end_index: 33
31	Lambda	parameter: 37 body: 34
32	Literal
33	Variable
34	IndexedVar	variable: 35 index: 37
35	Variable
36	ExprTuple	37
37	Variable