Expression of type Forall

from the theory of proveit.numbers.exponentiation

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import ExprRange, IndexedVar, Variable, a, b, m
from proveit.core_expr_types import a_1_to_m
from proveit.logic import Equals, Forall, NotEquals
from proveit.numbers import Complex, Exp, Mult, Real, one, zero
from proveit.numbers.exponentiation import prod_ai_raise_b__1_to_m

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [a_1_to_m, b], instance_expr = Equals(Exp(Mult(a_1_to_m), b), prod_ai_raise_b__1_to_m), domains = [Complex, Real], condition = ExprRange(sub_expr1, NotEquals(IndexedVar(a, sub_expr1), zero), one, m))

expr:

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

Passed sanity check: expr matches stored_expr

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

\forall_{\left(a_{1} \in \mathbb{C}\right), \left(a_{2} \in \mathbb{C}\right), \ldots, \left(a_{m} \in \mathbb{C}\right), b \in \mathbb{R}~|~\left(a_{1} \neq 0\right), \left(a_{2} \neq 0\right), \ldots, \left(a_{m} \neq 0\right)}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{m}\right)^{b} = \left(\left(a_{1}\right)^{b} \cdot  \left(a_{2}\right)^{b} \cdot  \ldots \cdot  \left(a_{m}\right)^{b}\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	34, 44
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15, 16
12	Operation	operator: 41 operands: 17
13	Operation	operator: 27 operands: 18
14	ExprRange	lambda_map: 19 start_index: 39 end_index: 40
15	Operation	operator: 30 operands: 20
16	ExprRange	lambda_map: 21 start_index: 39 end_index: 40
17	ExprTuple	22, 44
18	ExprTuple	23
19	Lambda	parameter: 47 body: 24
20	ExprTuple	44, 25
21	Lambda	parameter: 47 body: 26
22	Operation	operator: 27 operands: 28
23	ExprRange	lambda_map: 29 start_index: 39 end_index: 40
24	Operation	operator: 30 operands: 31
25	Literal
26	Operation	operator: 32 operands: 33
27	Literal
28	ExprTuple	34
29	Lambda	parameter: 47 body: 35
30	Literal
31	ExprTuple	43, 36
32	Literal
33	ExprTuple	43, 37
34	ExprRange	lambda_map: 38 start_index: 39 end_index: 40
35	Operation	operator: 41 operands: 42
36	Literal
37	Literal
38	Lambda	parameter: 47 body: 43
39	Literal
40	Variable
41	Literal
42	ExprTuple	43, 44
43	IndexedVar	variable: 45 index: 47
44	Variable
45	Variable
46	ExprTuple	47
47	Variable