Step |
Hyp |
Ref |
Expression |
1 |
|
sumsplit.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
sumsplit.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
sumsplit.3 |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
4 |
|
sumsplit.4 |
⊢ ( 𝜑 → ( 𝐴 ∪ 𝐵 ) ⊆ 𝑍 ) |
5 |
|
sumsplit.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ) |
6 |
|
sumsplit.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) |
7 |
|
sumsplit.7 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ) → 𝐶 ∈ ℂ ) |
8 |
|
sumsplit.8 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
9 |
|
sumsplit.9 |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐺 ) ∈ dom ⇝ ) |
10 |
7
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 ∈ ℂ ) |
11 |
1
|
eqimssi |
⊢ 𝑍 ⊆ ( ℤ≥ ‘ 𝑀 ) |
12 |
11
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
13 |
12
|
orcd |
⊢ ( 𝜑 → ( 𝑍 ⊆ ( ℤ≥ ‘ 𝑀 ) ∨ 𝑍 ∈ Fin ) ) |
14 |
|
sumss2 |
⊢ ( ( ( ( 𝐴 ∪ 𝐵 ) ⊆ 𝑍 ∧ ∀ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 ∈ ℂ ) ∧ ( 𝑍 ⊆ ( ℤ≥ ‘ 𝑀 ) ∨ 𝑍 ∈ Fin ) ) → Σ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) ) |
15 |
4 10 13 14
|
syl21anc |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) ) |
16 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) = 𝐶 ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) = 𝐶 ) |
18 |
|
elun1 |
⊢ ( 𝑘 ∈ 𝐴 → 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ) |
19 |
18 7
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
20 |
17 19
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ∈ ℂ ) |
21 |
|
iffalse |
⊢ ( ¬ 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) = 0 ) |
22 |
|
0cn |
⊢ 0 ∈ ℂ |
23 |
21 22
|
eqeltrdi |
⊢ ( ¬ 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ∈ ℂ ) |
24 |
23
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ∈ ℂ ) |
25 |
20 24
|
pm2.61dan |
⊢ ( 𝜑 → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ∈ ℂ ) |
26 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ∈ ℂ ) |
27 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝐵 → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) = 𝐶 ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) = 𝐶 ) |
29 |
|
elun2 |
⊢ ( 𝑘 ∈ 𝐵 → 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ) |
30 |
29 7
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝐶 ∈ ℂ ) |
31 |
28 30
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ∈ ℂ ) |
32 |
|
iffalse |
⊢ ( ¬ 𝑘 ∈ 𝐵 → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) = 0 ) |
33 |
32 22
|
eqeltrdi |
⊢ ( ¬ 𝑘 ∈ 𝐵 → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ∈ ℂ ) |
34 |
33
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ∈ ℂ ) |
35 |
31 34
|
pm2.61dan |
⊢ ( 𝜑 → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ∈ ℂ ) |
36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ∈ ℂ ) |
37 |
1 2 5 26 6 36 8 9
|
isumadd |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝑍 ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) = ( Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
38 |
19
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐶 + 0 ) = 𝐶 ) |
39 |
|
noel |
⊢ ¬ 𝑘 ∈ ∅ |
40 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑘 ∈ ( 𝐴 ∩ 𝐵 ) ↔ 𝑘 ∈ ∅ ) ) |
41 |
|
elin |
⊢ ( 𝑘 ∈ ( 𝐴 ∩ 𝐵 ) ↔ ( 𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) |
42 |
40 41
|
bitr3di |
⊢ ( 𝜑 → ( 𝑘 ∈ ∅ ↔ ( 𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) ) |
43 |
39 42
|
mtbii |
⊢ ( 𝜑 → ¬ ( 𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) |
44 |
|
imnan |
⊢ ( ( 𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵 ) ↔ ¬ ( 𝑘 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵 ) ) |
45 |
43 44
|
sylibr |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵 ) ) |
46 |
45
|
imp |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ¬ 𝑘 ∈ 𝐵 ) |
47 |
46 32
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) = 0 ) |
48 |
17 47
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) = ( 𝐶 + 0 ) ) |
49 |
|
iftrue |
⊢ ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = 𝐶 ) |
50 |
18 49
|
syl |
⊢ ( 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = 𝐶 ) |
51 |
50
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = 𝐶 ) |
52 |
38 48 51
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
53 |
35
|
addid2d |
⊢ ( 𝜑 → ( 0 + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) = if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) |
54 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → ( 0 + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) = if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) |
55 |
21
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) = 0 ) |
56 |
55
|
oveq1d |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) = ( 0 + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
57 |
|
elun |
⊢ ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) |
58 |
|
biorf |
⊢ ( ¬ 𝑘 ∈ 𝐴 → ( 𝑘 ∈ 𝐵 ↔ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) ) |
59 |
57 58
|
bitr4id |
⊢ ( ¬ 𝑘 ∈ 𝐴 → ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ↔ 𝑘 ∈ 𝐵 ) ) |
60 |
59
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ↔ 𝑘 ∈ 𝐵 ) ) |
61 |
60
|
ifbid |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) |
62 |
54 56 61
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ ¬ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
63 |
52 62
|
pm2.61dan |
⊢ ( 𝜑 → if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
64 |
63
|
sumeq2sdv |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) = Σ 𝑘 ∈ 𝑍 ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
65 |
4
|
unssad |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑍 ) |
66 |
19
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐶 ∈ ℂ ) |
67 |
|
sumss2 |
⊢ ( ( ( 𝐴 ⊆ 𝑍 ∧ ∀ 𝑘 ∈ 𝐴 𝐶 ∈ ℂ ) ∧ ( 𝑍 ⊆ ( ℤ≥ ‘ 𝑀 ) ∨ 𝑍 ∈ Fin ) ) → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ) |
68 |
65 66 13 67
|
syl21anc |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐶 = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) ) |
69 |
4
|
unssbd |
⊢ ( 𝜑 → 𝐵 ⊆ 𝑍 ) |
70 |
30
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐵 𝐶 ∈ ℂ ) |
71 |
|
sumss2 |
⊢ ( ( ( 𝐵 ⊆ 𝑍 ∧ ∀ 𝑘 ∈ 𝐵 𝐶 ∈ ℂ ) ∧ ( 𝑍 ⊆ ( ℤ≥ ‘ 𝑀 ) ∨ 𝑍 ∈ Fin ) ) → Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) |
72 |
69 70 13 71
|
syl21anc |
⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐵 𝐶 = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) |
73 |
68 72
|
oveq12d |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐶 + Σ 𝑘 ∈ 𝐵 𝐶 ) = ( Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐴 , 𝐶 , 0 ) + Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ 𝐵 , 𝐶 , 0 ) ) ) |
74 |
37 64 73
|
3eqtr4rd |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐶 + Σ 𝑘 ∈ 𝐵 𝐶 ) = Σ 𝑘 ∈ 𝑍 if ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) , 𝐶 , 0 ) ) |
75 |
15 74
|
eqtr4d |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) 𝐶 = ( Σ 𝑘 ∈ 𝐴 𝐶 + Σ 𝑘 ∈ 𝐵 𝐶 ) ) |