Expression of type Iff

from the theory of proveit.linear_algebra.linear_maps

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, K, P, Px, Py, V, W, c, x, y
from proveit.linear_algebra import LinMap, ScalarMult, VecAdd
from proveit.logic import And, Equals, Forall, Iff, InSet

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = Iff(InSet(P, LinMap(V, W)), And(Forall(instance_param_or_params = sub_expr1, instance_expr = InSet(Px, W), domain = V), Forall(instance_param_or_params = [x, y], instance_expr = Equals(Function(P, [VecAdd(x, y)]), VecAdd(Px, Py)), domain = V), Forall(instance_param_or_params = [c], instance_expr = Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Function(P, [ScalarMult(c, x)]), ScalarMult(c, Px)), domain = V), domain = K)).with_wrapping_at(2,4)).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 \in \mathcal{L}\left(V, W\right)\right) \Leftrightarrow  \\ \left(\begin{array}{c} \left[\forall_{x \in V}~\left(P\left(x\right) \in W\right)\right] \land  \\ \left[\forall_{x, y \in V}~\left(P\left(x + y\right) = \left(P\left(x\right) + P\left(y\right)\right)\right)\right] \land  \\ \left[\forall_{c \in K}~\left[\forall_{x \in V}~\left(P\left(c \cdot x\right) = \left(c \cdot P\left(x\right)\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: 54 operands: 5
4	Operation	operator: 30 operands: 6
5	ExprTuple	66, 7
6	ExprTuple	8, 9, 10
7	Operation	operator: 11 operands: 12
8	Operation	operator: 32 operand: 16
9	Operation	operator: 32 operand: 17
10	Operation	operator: 32 operand: 18
11	Literal
12	ExprTuple	59, 35
13	ExprTuple	16
14	ExprTuple	17
15	ExprTuple	18
16	Lambda	parameter: 69 body: 19
17	Lambda	parameters: 50 body: 20
18	Lambda	parameter: 68 body: 22
19	Conditional	value: 23 condition: 48
20	Conditional	value: 24 condition: 25
21	ExprTuple	68
22	Conditional	value: 26 condition: 27
23	Operation	operator: 54 operands: 28
24	Operation	operator: 52 operands: 29
25	Operation	operator: 30 operands: 31
26	Operation	operator: 32 operand: 39
27	Operation	operator: 54 operands: 34
28	ExprTuple	63, 35
29	ExprTuple	36, 37
30	Literal
31	ExprTuple	48, 38
32	Literal
33	ExprTuple	39
34	ExprTuple	68, 40
35	Variable
36	Operation	operator: 66 operand: 45
37	Operation	operator: 49 operands: 42
38	Operation	operator: 54 operands: 43
39	Lambda	parameter: 69 body: 44
40	Variable
41	ExprTuple	45
42	ExprTuple	63, 46
43	ExprTuple	56, 59
44	Conditional	value: 47 condition: 48
45	Operation	operator: 49 operands: 50
46	Operation	operator: 66 operand: 56
47	Operation	operator: 52 operands: 53
48	Operation	operator: 54 operands: 55
49	Literal
50	ExprTuple	69, 56
51	ExprTuple	56
52	Literal
53	ExprTuple	57, 58
54	Literal
55	ExprTuple	69, 59
56	Variable
57	Operation	operator: 66 operand: 62
58	Operation	operator: 64 operands: 61
59	Variable
60	ExprTuple	62
61	ExprTuple	68, 63
62	Operation	operator: 64 operands: 65
63	Operation	operator: 66 operand: 69
64	Literal
65	ExprTuple	68, 69
66	Variable
67	ExprTuple	69
68	Variable
69	Variable