Expression of type Implies

from the theory of proveit.numbers.summation

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 Function, R, S, a, f, fa, fx, gy, x, y
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import Complex, Sum

# build up the expression from sub-expressions
expr = Implies(Forall(instance_param_or_params = [a], instance_expr = InSet(fa, Complex), domain = R), Equals(Sum(index_or_indices = [x], summand = fx, domain = R), Sum(index_or_indices = [y], summand = Function(f, [gy]), domain = S)))

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_{a \in R}~\left(f\left(a\right) \in \mathbb{C}\right)\right] \Rightarrow \left(\left[\sum_{x \in R}~f\left(x\right)\right] = \left[\sum_{y \in S}~f\left(g\left(y\right)\right)\right]\right)

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.	()	()	('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: 9
4	Operation	operator: 7 operands: 8
5	Literal
6	ExprTuple	9
7	Literal
8	ExprTuple	10, 11
9	Lambda	parameter: 37 body: 12
10	Operation	operator: 14 operand: 18
11	Operation	operator: 14 operand: 19
12	Conditional	value: 16 condition: 17
13	ExprTuple	18
14	Literal
15	ExprTuple	19
16	Operation	operator: 35 operands: 20
17	Operation	operator: 35 operands: 21
18	Lambda	parameter: 38 body: 22
19	Lambda	parameter: 44 body: 23
20	ExprTuple	24, 25
21	ExprTuple	37, 39
22	Conditional	value: 26 condition: 27
23	Conditional	value: 28 condition: 29
24	Operation	operator: 33 operand: 37
25	Literal
26	Operation	operator: 33 operand: 38
27	Operation	operator: 35 operands: 32
28	Operation	operator: 33 operand: 40
29	Operation	operator: 35 operands: 36
30	ExprTuple	37
31	ExprTuple	38
32	ExprTuple	38, 39
33	Variable
34	ExprTuple	40
35	Literal
36	ExprTuple	44, 41
37	Variable
38	Variable
39	Variable
40	Operation	operator: 42 operand: 44
41	Variable
42	Variable
43	ExprTuple	44
44	Variable