Theorem euclids_lemma of type Forall

from the theory of proveit.numbers.divisibility

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.numbers.divisibility import euclids_lemma

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

Passed sanity check: euclids_lemma matches stored_expr

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

\forall_{x, y, z \in \mathbb{Z}~|~x \rvert \left(y \cdot z\right), gcd\left(x, y\right) = 1}~\left(x \rvert z\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	33, 34, 32
5	Conditional	value: 6 condition: 7
6	Operation	operator: 20 operands: 8
7	Operation	operator: 9 operands: 10
8	ExprTuple	33, 32
9	Literal
10	ExprTuple	11, 12, 13, 14, 15
11	Operation	operator: 18 operands: 16
12	Operation	operator: 18 operands: 17
13	Operation	operator: 18 operands: 19
14	Operation	operator: 20 operands: 21
15	Operation	operator: 22 operands: 23
16	ExprTuple	33, 24
17	ExprTuple	34, 24
18	Literal
19	ExprTuple	32, 24
20	Literal
21	ExprTuple	33, 25
22	Literal
23	ExprTuple	26, 27
24	Literal
25	Operation	operator: 28 operands: 29
26	Operation	operator: 30 operands: 31
27	Literal
28	Literal
29	ExprTuple	34, 32
30	Literal
31	ExprTuple	33, 34
32	Variable
33	Variable
34	Variable