Expression of type Conditional

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 Conditional, ExprRange, IndexedVar, V, Variable, n, x
from proveit.core_expr_types import x_1_to_n
from proveit.linear_algebra import Span, SpanningSets
from proveit.logic import And, Equals, InSet, Set
from proveit.numbers import one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
sub_expr2 = Set(x_1_to_n)
expr = Conditional(Equals(InSet(sub_expr2, SpanningSets(V)), Equals(V, Span(sub_expr2))), And(ExprRange(sub_expr1, InSet(IndexedVar(x, sub_expr1), V), one, n)))

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\{\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{SpanningSets}\left(V\right)\right) = \left(V = \textrm{Span}\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right)\right) \textrm{ if } \left(x_{1} \in V\right) \land  \left(x_{2} \in V\right) \land  \ldots \land  \left(x_{n} \in V\right)\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 10 operands: 3
2	Operation	operator: 4 operands: 5
3	ExprTuple	6, 7
4	Literal
5	ExprTuple	8
6	Operation	operator: 20 operands: 9
7	Operation	operator: 10 operands: 11
8	ExprRange	lambda_map: 12 start_index: 28 end_index: 29
9	ExprTuple	22, 13
10	Literal
11	ExprTuple	23, 14
12	Lambda	parameter: 33 body: 15
13	Operation	operator: 16 operand: 23
14	Operation	operator: 18 operand: 22
15	Operation	operator: 20 operands: 21
16	Literal
17	ExprTuple	23
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	30, 23
22	Operation	operator: 24 operands: 25
23	Variable
24	Literal
25	ExprTuple	26
26	ExprRange	lambda_map: 27 start_index: 28 end_index: 29
27	Lambda	parameter: 33 body: 30
28	Literal
29	Variable
30	IndexedVar	variable: 31 index: 33
31	Variable
32	ExprTuple	33
33	Variable