Theorem ortho_norm_basis_existence_with_any_vector of type Forall

from the theory of proveit.linear_algebra.inner_products

see dependencies

import proveit
# Automation is not needed when only building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_theorem_expr # Load the stored theorem expression as 'stored_expr'
# import the special expression
from proveit.linear_algebra.inner_products import ortho_norm_basis_existence_with_any_vector

# check that the built expression is the same as the stored expression
assert ortho_norm_basis_existence_with_any_vector.expr == stored_expr
assert ortho_norm_basis_existence_with_any_vector.expr._style_id == stored_expr._style_id
print("Passed sanity check: ortho_norm_basis_existence_with_any_vector matches stored_expr")

Passed sanity check: ortho_norm_basis_existence_with_any_vector matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{K}~\left[\forall_{n \in \mathbb{N}^+}~\left[\begin{array}{l}\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)~|~\textrm{Dim}\left(H\right) = n}~\\
\left[\forall_{v \in H}~\left[\exists_{x_{1}, x_{2}, \ldots, x_{n}}~\left(\begin{array}{c} \left(v \in \left\{x_{1}, x_{2}, \ldots, x_{n}\right\}\right) \land  \\ \left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(H\right)\right) \end{array}\right)\right]\right]\end{array}\right]\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 17 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 40 body: 3
3	Operation	operator: 17 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 60 body: 7
6	ExprTuple	60
7	Conditional	value: 8 condition: 9
8	Operation	operator: 17 operand: 12
9	Operation	operator: 47 operands: 11
10	ExprTuple	12
11	ExprTuple	60, 13
12	Lambda	parameter: 57 body: 14
13	Literal
14	Conditional	value: 15 condition: 16
15	Operation	operator: 17 operand: 20
16	Operation	operator: 42 operands: 19
17	Literal
18	ExprTuple	20
19	ExprTuple	21, 22
20	Lambda	parameter: 49 body: 24
21	Operation	operator: 25 operands: 26
22	Operation	operator: 27 operands: 28
23	ExprTuple	49
24	Conditional	value: 29 condition: 30
25	Literal
26	ExprTuple	57, 31
27	Literal
28	ExprTuple	32, 60
29	Operation	operator: 33 operand: 39
30	Operation	operator: 47 operands: 35
31	Operation	operator: 36 operand: 40
32	Operation	operator: 38 operand: 57
33	Literal
34	ExprTuple	39
35	ExprTuple	49, 57
36	Literal
37	ExprTuple	40
38	Literal
39	Lambda	parameters: 53 body: 41
40	Variable
41	Operation	operator: 42 operands: 43
42	Literal
43	ExprTuple	44, 45
44	Operation	operator: 47 operands: 46
45	Operation	operator: 47 operands: 48
46	ExprTuple	49, 50
47	Literal
48	ExprTuple	50, 51
49	Variable
50	Operation	operator: 52 operands: 53
51	Operation	operator: 54 operand: 57
52	Literal
53	ExprTuple	56
54	Literal
55	ExprTuple	57
56	ExprRange	lambda_map: 58 start_index: 59 end_index: 60
57	Variable
58	Lambda	parameter: 64 body: 61
59	Literal
60	Variable
61	IndexedVar	variable: 62 index: 64
62	Variable
63	ExprTuple	64
64	Variable