Expression of type Equals

from the theory of proveit.linear_algebra.vector_sets

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 K, V
from proveit.core_expr_types import a_1_to_n, x_1_to_n
from proveit.linear_algebra import LinDepSets, VecZero, lin_comb_axn, some_nonzero_a
from proveit.logic import Equals, Exists, InSet, Set

# build up the expression from sub-expressions
expr = Equals(InSet(Set(x_1_to_n), LinDepSets(V)), Exists(instance_param_or_params = [a_1_to_n], instance_expr = Equals(lin_comb_axn, VecZero(V)), domain = K, condition = some_nonzero_a).with_wrapping()).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\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{LinDepSets}\left(V\right)\right) =  \\ \left[\begin{array}{l}\exists_{a_{1}, a_{2}, \ldots, a_{n} \in K~|~\left(a_{1} \neq 0\right) \lor  \left(a_{2} \neq 0\right) \lor  \ldots \lor  \left(a_{n} \neq 0\right)}~\\
\left(\left(\left(a_{1} \cdot x_{1}\right) +  \left(a_{2} \cdot x_{2}\right) +  \ldots +  \left(a_{n} \cdot x_{n}\right)\right) = \vec{0}\left(V\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: 21 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 41 operands: 4
3	Operation	operator: 5 operand: 9
4	ExprTuple	7, 8
5	Literal
6	ExprTuple	9
7	Operation	operator: 10 operands: 11
8	Operation	operator: 12 operand: 37
9	Lambda	parameters: 13 body: 14
10	Literal
11	ExprTuple	15
12	Literal
13	ExprTuple	16
14	Conditional	value: 17 condition: 18
15	ExprRange	lambda_map: 19 start_index: 44 end_index: 45
16	ExprRange	lambda_map: 20 start_index: 44 end_index: 45
17	Operation	operator: 21 operands: 22
18	Operation	operator: 23 operands: 24
19	Lambda	parameter: 59 body: 53
20	Lambda	parameter: 59 body: 54
21	Literal
22	ExprTuple	25, 26
23	Literal
24	ExprTuple	27, 28
25	Operation	operator: 29 operands: 30
26	Operation	operator: 31 operand: 37
27	ExprRange	lambda_map: 33 start_index: 44 end_index: 45
28	Operation	operator: 34 operands: 35
29	Literal
30	ExprTuple	36
31	Literal
32	ExprTuple	37
33	Lambda	parameter: 59 body: 38
34	Literal
35	ExprTuple	39
36	ExprRange	lambda_map: 40 start_index: 44 end_index: 45
37	Variable
38	Operation	operator: 41 operands: 42
39	ExprRange	lambda_map: 43 start_index: 44 end_index: 45
40	Lambda	parameter: 59 body: 46
41	Literal
42	ExprTuple	54, 47
43	Lambda	parameter: 59 body: 48
44	Literal
45	Variable
46	Operation	operator: 49 operands: 50
47	Variable
48	Operation	operator: 51 operands: 52
49	Literal
50	ExprTuple	54, 53
51	Literal
52	ExprTuple	54, 55
53	IndexedVar	variable: 56 index: 59
54	IndexedVar	variable: 57 index: 59
55	Literal
56	Variable
57	Variable
58	ExprTuple	59
59	Variable