Expression of type Conditional

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 Conditional, ExprRange, IndexedVar, Variable, a, b, m
from proveit.core_expr_types import a_1_to_m
from proveit.logic import Equals, Forall, InSet, NotEquals
from proveit.numbers import Complex, Exp, Mult, NaturalPos, 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 = Conditional(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)), InSet(m, NaturalPos))

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())

\left\{\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) \textrm{ if } m \in \mathbb{N}^+\right..

stored_expr.style_options()

name	description	default	current value	related methods
condition_delimiter	'comma' or 'and'	comma	comma	('with_comma_delimiter', 'with_conjunction_delimiter')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Conditional	value: 1 condition: 2
1	Operation	operator: 3 operand: 6
2	Operation	operator: 34 operands: 5
3	Literal
4	ExprTuple	6
5	ExprTuple	44, 7
6	Lambda	parameters: 8 body: 9
7	Literal
8	ExprTuple	38, 48
9	Conditional	value: 10 condition: 11
10	Operation	operator: 12 operands: 13
11	Operation	operator: 14 operands: 15
12	Literal
13	ExprTuple	16, 17
14	Literal
15	ExprTuple	18, 19, 20
16	Operation	operator: 45 operands: 21
17	Operation	operator: 31 operands: 22
18	ExprRange	lambda_map: 23 start_index: 43 end_index: 44
19	Operation	operator: 34 operands: 24
20	ExprRange	lambda_map: 25 start_index: 43 end_index: 44
21	ExprTuple	26, 48
22	ExprTuple	27
23	Lambda	parameter: 51 body: 28
24	ExprTuple	48, 29
25	Lambda	parameter: 51 body: 30
26	Operation	operator: 31 operands: 32
27	ExprRange	lambda_map: 33 start_index: 43 end_index: 44
28	Operation	operator: 34 operands: 35
29	Literal
30	Operation	operator: 36 operands: 37
31	Literal
32	ExprTuple	38
33	Lambda	parameter: 51 body: 39
34	Literal
35	ExprTuple	47, 40
36	Literal
37	ExprTuple	47, 41
38	ExprRange	lambda_map: 42 start_index: 43 end_index: 44
39	Operation	operator: 45 operands: 46
40	Literal
41	Literal
42	Lambda	parameter: 51 body: 47
43	Literal
44	Variable
45	Literal
46	ExprTuple	47, 48
47	IndexedVar	variable: 49 index: 51
48	Variable
49	Variable
50	ExprTuple	51
51	Variable