Theorem cauchy_schwartz_equality 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 cauchy_schwartz_equality

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

Passed sanity check: cauchy_schwartz_equality matches stored_expr

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

\forall_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(\mathbb{C}\right)}~\left[\forall_{v, w \in H}~\left(\left(\left \|\left \langle v, w\right \rangle\right \|^{2} = \left(\left \langle v, v\right \rangle \cdot \left \langle w, w\right \rangle\right)\right) \Leftrightarrow \left(v \in \textrm{Span}\left(\left\{w\right\}\right)\right)\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 36 body: 4
3	ExprTuple	36
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	36, 12
11	Lambda	parameters: 56 body: 13
12	Operation	operator: 14 operand: 18
13	Conditional	value: 16 condition: 17
14	Literal
15	ExprTuple	18
16	Operation	operator: 19 operands: 20
17	Operation	operator: 21 operands: 22
18	Literal
19	Literal
20	ExprTuple	23, 24
21	Literal
22	ExprTuple	25, 26
23	Operation	operator: 27 operands: 28
24	Operation	operator: 31 operands: 29
25	Operation	operator: 31 operands: 30
26	Operation	operator: 31 operands: 32
27	Literal
28	ExprTuple	33, 34
29	ExprTuple	57, 35
30	ExprTuple	57, 36
31	Literal
32	ExprTuple	58, 36
33	Operation	operator: 37 operands: 38
34	Operation	operator: 39 operands: 40
35	Operation	operator: 41 operand: 47
36	Variable
37	Literal
38	ExprTuple	43, 44
39	Literal
40	ExprTuple	45, 46
41	Literal
42	ExprTuple	47
43	Operation	operator: 48 operand: 54
44	Literal
45	Operation	operator: 55 operands: 50
46	Operation	operator: 55 operands: 51
47	Operation	operator: 52 operand: 58
48	Literal
49	ExprTuple	54
50	ExprTuple	57, 57
51	ExprTuple	58, 58
52	Literal
53	ExprTuple	58
54	Operation	operator: 55 operands: 56
55	Literal
56	ExprTuple	57, 58
57	Variable
58	Variable