Expression of type Iff

from the theory of proveit.linear_algebra.inner_products

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 P, Px, Py, X, x, y
from proveit.linear_algebra import Hspace, InnerProd, OrthoProj, VecZero
from proveit.logic import And, Equals, Forall, Iff, Implies
from proveit.numbers import zero

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Iff(Equals(P, OrthoProj(Hspace, X)), And(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Px, x), domain = X), Forall(instance_param_or_params = [y], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), domain = Hspace)).with_wrapping_at(2)).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(P = \textrm{OrthoProj}\left(\mathcal{H}, X\right)\right) \Leftrightarrow  \\ \left(\begin{array}{c} \left[\forall_{x \in X}~\left(P\left(x\right) = x\right)\right] \land  \\ \left[\forall_{y \in \mathcal{H}}~\left(\left[\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right)\right] \Rightarrow \left(P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right)\right)\right] \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: 44 operands: 5
4	Operation	operator: 6 operands: 7
5	ExprTuple	37, 8
6	Literal
7	ExprTuple	9, 10
8	Operation	operator: 11 operands: 12
9	Operation	operator: 29 operand: 15
10	Operation	operator: 29 operand: 16
11	Literal
12	ExprTuple	43, 50
13	ExprTuple	15
14	ExprTuple	16
15	Lambda	parameter: 53 body: 17
16	Lambda	parameter: 54 body: 18
17	Conditional	value: 19 condition: 42
18	Conditional	value: 20 condition: 21
19	Operation	operator: 44 operands: 22
20	Operation	operator: 23 operands: 24
21	Operation	operator: 46 operands: 25
22	ExprTuple	26, 53
23	Literal
24	ExprTuple	27, 28
25	ExprTuple	54, 43
26	Operation	operator: 37 operand: 53
27	Operation	operator: 29 operand: 32
28	Operation	operator: 44 operands: 31
29	Literal
30	ExprTuple	32
31	ExprTuple	33, 34
32	Lambda	parameter: 53 body: 36
33	Operation	operator: 37 operand: 54
34	Operation	operator: 39 operand: 43
35	ExprTuple	53
36	Conditional	value: 41 condition: 42
37	Variable
38	ExprTuple	54
39	Literal
40	ExprTuple	43
41	Operation	operator: 44 operands: 45
42	Operation	operator: 46 operands: 47
43	Variable
44	Literal
45	ExprTuple	48, 49
46	Literal
47	ExprTuple	53, 50
48	Operation	operator: 51 operands: 52
49	Literal
50	Variable
51	Literal
52	ExprTuple	53, 54
53	Variable
54	Variable