Theorem true_for_each_is_true_for_all of type Forall

from the theory of proveit.logic.sets.enumeration

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.logic.sets.enumeration import true_for_each_is_true_for_all

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

Passed sanity check: true_for_each_is_true_for_all matches stored_expr

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

\forall_{n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{n}}~\left[\forall_{P}~\left(\left(P\left(a_{1}\right) \land  P\left(a_{2}\right) \land  \ldots \land  P\left(a_{n}\right)\right) = \left[\forall_{x \in \left\{a_{1}, a_{2}, \ldots, a_{n}\right\}}~P\left(x\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: 22 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 43 body: 4
3	ExprTuple	43
4	Conditional	value: 5 condition: 6
5	Operation	operator: 22 operand: 9
6	Operation	operator: 34 operands: 8
7	ExprTuple	9
8	ExprTuple	43, 10
9	Lambda	parameters: 39 body: 11
10	Literal
11	Operation	operator: 22 operand: 13
12	ExprTuple	13
13	Lambda	parameter: 32 body: 15
14	ExprTuple	32
15	Operation	operator: 16 operands: 17
16	Literal
17	ExprTuple	18, 19
18	Operation	operator: 20 operands: 21
19	Operation	operator: 22 operand: 25
20	Literal
21	ExprTuple	24
22	Literal
23	ExprTuple	25
24	ExprRange	lambda_map: 26 start_index: 42 end_index: 43
25	Lambda	parameter: 36 body: 27
26	Lambda	parameter: 47 body: 28
27	Conditional	value: 29 condition: 30
28	Operation	operator: 32 operand: 44
29	Operation	operator: 32 operand: 36
30	Operation	operator: 34 operands: 35
31	ExprTuple	44
32	Variable
33	ExprTuple	36
34	Literal
35	ExprTuple	36, 37
36	Variable
37	Operation	operator: 38 operands: 39
38	Literal
39	ExprTuple	40
40	ExprRange	lambda_map: 41 start_index: 42 end_index: 43
41	Lambda	parameter: 47 body: 44
42	Literal
43	Variable
44	IndexedVar	variable: 45 index: 47
45	Variable
46	ExprTuple	47
47	Variable