Theorem ortho_norm_basis_existence 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

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

Passed sanity check: ortho_norm_basis_existence 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[\exists_{x_{1}, x_{2}, \ldots, x_{n}}~\left(\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \in \textrm{O.N.Bases}\left(H\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: 10 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 43 body: 3
3	Operation	operator: 10 operand: 5
4	ExprTuple	5
5	Lambda	parameter: 48 body: 7
6	ExprTuple	48
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operand: 13
9	Operation	operator: 30 operands: 12
10	Literal
11	ExprTuple	13
12	ExprTuple	48, 14
13	Lambda	parameter: 45 body: 15
14	Literal
15	Conditional	value: 16 condition: 17
16	Operation	operator: 18 operand: 22
17	Operation	operator: 20 operands: 21
18	Literal
19	ExprTuple	22
20	Literal
21	ExprTuple	23, 24
22	Lambda	parameters: 40 body: 25
23	Operation	operator: 26 operands: 27
24	Operation	operator: 28 operands: 29
25	Operation	operator: 30 operands: 31
26	Literal
27	ExprTuple	45, 32
28	Literal
29	ExprTuple	33, 48
30	Literal
31	ExprTuple	34, 35
32	Operation	operator: 36 operand: 43
33	Operation	operator: 38 operand: 45
34	Operation	operator: 39 operands: 40
35	Operation	operator: 41 operand: 45
36	Literal
37	ExprTuple	43
38	Literal
39	Literal
40	ExprTuple	44
41	Literal
42	ExprTuple	45
43	Variable
44	ExprRange	lambda_map: 46 start_index: 47 end_index: 48
45	Variable
46	Lambda	parameter: 52 body: 49
47	Literal
48	Variable
49	IndexedVar	variable: 50 index: 52
50	Variable
51	ExprTuple	52
52	Variable