Theorem distribute_through_abs_sum of type Forall

from the theory of proveit.numbers.multiplication

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.multiplication import distribute_through_abs_sum

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

Passed sanity check: distribute_through_abs_sum matches stored_expr

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

\forall_{i, j, k \in \mathbb{N}}~\left[\begin{array}{l}\forall_{a_{1}, a_{2}, \ldots, a_{i}, c_{1}, c_{2}, \ldots, c_{k} \in \mathbb{R}^{\ge 0}}~\\
\left[\begin{array}{l}\forall_{b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\\
\left(\begin{array}{c} \begin{array}{l} \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot \left|b_{1} +  b_{2} +  \ldots +  b_{j}\right|\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) =  \\ \left|\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{1}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{2}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right) +  \ldots +  \left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot b_{j}\cdot c_{1} \cdot  c_{2} \cdot  \ldots \cdot  c_{k}\right)\right| \end{array} \end{array}\right)\end{array}\right]\end{array}\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: 21 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	72, 64, 77
4	Conditional	value: 5 condition: 6
5	Operation	operator: 21 operand: 9
6	Operation	operator: 36 operands: 8
7	ExprTuple	9
8	ExprTuple	10, 11, 12
9	Lambda	parameters: 13 body: 14
10	Operation	operator: 53 operands: 15
11	Operation	operator: 53 operands: 16
12	Operation	operator: 53 operands: 17
13	ExprTuple	68, 70
14	Conditional	value: 18 condition: 19
15	ExprTuple	72, 20
16	ExprTuple	64, 20
17	ExprTuple	77, 20
18	Operation	operator: 21 operand: 24
19	Operation	operator: 36 operands: 23
20	Literal
21	Literal
22	ExprTuple	24
23	ExprTuple	25, 26
24	Lambda	parameters: 59 body: 27
25	ExprRange	lambda_map: 28 start_index: 76 end_index: 72
26	ExprRange	lambda_map: 29 start_index: 76 end_index: 77
27	Conditional	value: 30 condition: 31
28	Lambda	parameter: 84 body: 32
29	Lambda	parameter: 84 body: 33
30	Operation	operator: 34 operands: 35
31	Operation	operator: 36 operands: 37
32	Operation	operator: 53 operands: 38
33	Operation	operator: 53 operands: 39
34	Literal
35	ExprTuple	40, 41
36	Literal
37	ExprTuple	42
38	ExprTuple	78, 43
39	ExprTuple	80, 43
40	Operation	operator: 65 operands: 44
41	Operation	operator: 50 operand: 48
42	ExprRange	lambda_map: 46 start_index: 76 end_index: 64
43	Literal
44	ExprTuple	68, 47, 70
45	ExprTuple	48
46	Lambda	parameter: 84 body: 49
47	Operation	operator: 50 operand: 55
48	Operation	operator: 58 operands: 52
49	Operation	operator: 53 operands: 54
50	Literal
51	ExprTuple	55
52	ExprTuple	56
53	Literal
54	ExprTuple	67, 57
55	Operation	operator: 58 operands: 59
56	ExprRange	lambda_map: 60 start_index: 76 end_index: 64
57	Literal
58	Literal
59	ExprTuple	61
60	Lambda	parameter: 79 body: 62
61	ExprRange	lambda_map: 63 start_index: 76 end_index: 64
62	Operation	operator: 65 operands: 66
63	Lambda	parameter: 84 body: 67
64	Variable
65	Literal
66	ExprTuple	68, 69, 70
67	IndexedVar	variable: 73 index: 84
68	ExprRange	lambda_map: 71 start_index: 76 end_index: 72
69	IndexedVar	variable: 73 index: 79
70	ExprRange	lambda_map: 75 start_index: 76 end_index: 77
71	Lambda	parameter: 84 body: 78
72	Variable
73	Variable
74	ExprTuple	79
75	Lambda	parameter: 84 body: 80
76	Literal
77	Variable
78	IndexedVar	variable: 81 index: 84
79	Variable
80	IndexedVar	variable: 82 index: 84
81	Variable
82	Variable
83	ExprTuple	84
84	Variable