Expression of type Lambda

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, Lambda, 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 = Lambda(m, 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())

m \mapsto \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()

no style options

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Lambda	parameter: 46 body: 2
1	ExprTuple	46
2	Conditional	value: 3 condition: 4
3	Operation	operator: 5 operand: 8
4	Operation	operator: 36 operands: 7
5	Literal
6	ExprTuple	8
7	ExprTuple	46, 9
8	Lambda	parameters: 10 body: 11
9	Literal
10	ExprTuple	40, 50
11	Conditional	value: 12 condition: 13
12	Operation	operator: 14 operands: 15
13	Operation	operator: 16 operands: 17
14	Literal
15	ExprTuple	18, 19
16	Literal
17	ExprTuple	20, 21, 22
18	Operation	operator: 47 operands: 23
19	Operation	operator: 33 operands: 24
20	ExprRange	lambda_map: 25 start_index: 45 end_index: 46
21	Operation	operator: 36 operands: 26
22	ExprRange	lambda_map: 27 start_index: 45 end_index: 46
23	ExprTuple	28, 50
24	ExprTuple	29
25	Lambda	parameter: 53 body: 30
26	ExprTuple	50, 31
27	Lambda	parameter: 53 body: 32
28	Operation	operator: 33 operands: 34
29	ExprRange	lambda_map: 35 start_index: 45 end_index: 46
30	Operation	operator: 36 operands: 37
31	Literal
32	Operation	operator: 38 operands: 39
33	Literal
34	ExprTuple	40
35	Lambda	parameter: 53 body: 41
36	Literal
37	ExprTuple	49, 42
38	Literal
39	ExprTuple	49, 43
40	ExprRange	lambda_map: 44 start_index: 45 end_index: 46
41	Operation	operator: 47 operands: 48
42	Literal
43	Literal
44	Lambda	parameter: 53 body: 49
45	Literal
46	Variable
47	Literal
48	ExprTuple	49, 50
49	IndexedVar	variable: 51 index: 53
50	Variable
51	Variable
52	ExprTuple	53
53	Variable