Expression of type Implies

from the theory of proveit.numbers.division

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 Px, d, x, y, z
from proveit.logic import Equals, Forall, Implies, InSet, NotEquals
from proveit.numbers import Complex, Interval, Sum, frac, zero

# build up the expression from sub-expressions
sub_expr1 = [x]
sub_expr2 = Interval(y, z)
expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(Px, Complex), domain = sub_expr2), Forall(instance_param_or_params = [d], instance_expr = Equals(frac(Sum(index_or_indices = sub_expr1, summand = Px, domain = sub_expr2), d), Sum(index_or_indices = sub_expr1, summand = frac(Px, d), domain = sub_expr2)), domain = Complex, condition = NotEquals(d, zero)))

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_{x \in \{y~\ldotp \ldotp~z\}}~\left(P\left(x\right) \in \mathbb{C}\right)\right] \Rightarrow \left[\forall_{d \in \mathbb{C}~|~d \neq 0}~\left(\frac{\sum_{x = y}^{z} P\left(x\right)}{d} = \left(\sum_{x = y}^{z} \frac{P\left(x\right)}{d}\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: 6 operand: 8
4	Operation	operator: 6 operand: 9
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameter: 50 body: 10
9	Lambda	parameter: 44 body: 12
10	Conditional	value: 13 condition: 42
11	ExprTuple	44
12	Conditional	value: 14 condition: 15
13	Operation	operator: 45 operands: 16
14	Operation	operator: 17 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	43, 32
17	Literal
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	Operation	operator: 40 operands: 25
22	Operation	operator: 34 operand: 31
23	Operation	operator: 45 operands: 27
24	Operation	operator: 28 operands: 29
25	ExprTuple	30, 44
26	ExprTuple	31
27	ExprTuple	44, 32
28	Literal
29	ExprTuple	44, 33
30	Operation	operator: 34 operand: 37
31	Lambda	parameter: 50 body: 36
32	Literal
33	Literal
34	Literal
35	ExprTuple	37
36	Conditional	value: 38 condition: 42
37	Lambda	parameter: 50 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: 46
43	Operation	operator: 47 operand: 50
44	Variable
45	Literal
46	ExprTuple	50, 49
47	Variable
48	ExprTuple	50
49	Operation	operator: 51 operands: 52
50	Variable
51	Literal
52	ExprTuple	53, 54
53	Variable
54	Variable