Step |
Hyp |
Ref |
Expression |
1 |
|
fprodsplit.1 |
⊢ ( 𝜑 → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
2 |
|
fprodsplit.2 |
⊢ ( 𝜑 → 𝑈 = ( 𝐴 ∪ 𝐵 ) ) |
3 |
|
fprodsplit.3 |
⊢ ( 𝜑 → 𝑈 ∈ Fin ) |
4 |
|
fprodsplit.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → 𝐶 ∈ ℂ ) |
5 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = 𝐶 ) |
6 |
5
|
prodeq2i |
⊢ ∏ 𝑘 ∈ 𝐴 if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = ∏ 𝑘 ∈ 𝐴 𝐶 |
7 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ 𝐵 ) |
8 |
7 2
|
sseqtrrid |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑈 ) |
9 |
5
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = 𝐶 ) |
10 |
8
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝑈 ) |
11 |
10 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
12 |
9 11
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ∈ ℂ ) |
13 |
|
eldifn |
⊢ ( 𝑘 ∈ ( 𝑈 ∖ 𝐴 ) → ¬ 𝑘 ∈ 𝐴 ) |
14 |
13
|
iffalsed |
⊢ ( 𝑘 ∈ ( 𝑈 ∖ 𝐴 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = 1 ) |
15 |
14
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑈 ∖ 𝐴 ) ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = 1 ) |
16 |
8 12 15 3
|
fprodss |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) |
17 |
6 16
|
syl5eqr |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) |
18 |
|
iftrue |
⊢ ( 𝑘 ∈ 𝐵 → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = 𝐶 ) |
19 |
18
|
prodeq2i |
⊢ ∏ 𝑘 ∈ 𝐵 if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = ∏ 𝑘 ∈ 𝐵 𝐶 |
20 |
|
ssun2 |
⊢ 𝐵 ⊆ ( 𝐴 ∪ 𝐵 ) |
21 |
20 2
|
sseqtrrid |
⊢ ( 𝜑 → 𝐵 ⊆ 𝑈 ) |
22 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = 𝐶 ) |
23 |
21
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝑘 ∈ 𝑈 ) |
24 |
23 4
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → 𝐶 ∈ ℂ ) |
25 |
22 24
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ∈ ℂ ) |
26 |
|
eldifn |
⊢ ( 𝑘 ∈ ( 𝑈 ∖ 𝐵 ) → ¬ 𝑘 ∈ 𝐵 ) |
27 |
26
|
iffalsed |
⊢ ( 𝑘 ∈ ( 𝑈 ∖ 𝐵 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = 1 ) |
28 |
27
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑈 ∖ 𝐵 ) ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = 1 ) |
29 |
21 25 28 3
|
fprodss |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐵 if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) |
30 |
19 29
|
syl5eqr |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐵 𝐶 = ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) |
31 |
17 30
|
oveq12d |
⊢ ( 𝜑 → ( ∏ 𝑘 ∈ 𝐴 𝐶 · ∏ 𝑘 ∈ 𝐵 𝐶 ) = ( ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) |
32 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
33 |
|
ifcl |
⊢ ( ( 𝐶 ∈ ℂ ∧ 1 ∈ ℂ ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ∈ ℂ ) |
34 |
4 32 33
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ∈ ℂ ) |
35 |
|
ifcl |
⊢ ( ( 𝐶 ∈ ℂ ∧ 1 ∈ ℂ ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ∈ ℂ ) |
36 |
4 32 35
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ∈ ℂ ) |
37 |
3 34 36
|
fprodmul |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑈 ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = ( ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · ∏ 𝑘 ∈ 𝑈 if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) |
38 |
2
|
eleq2d |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑈 ↔ 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ) ) |
39 |
|
elun |
⊢ ( 𝑘 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) |
40 |
38 39
|
bitrdi |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑈 ↔ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) ) |
41 |
40
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) |
42 |
|
disjel |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) = ∅ ∧ 𝑘 ∈ 𝐴 ) → ¬ 𝑘 ∈ 𝐵 ) |
43 |
1 42
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ¬ 𝑘 ∈ 𝐵 ) |
44 |
43
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) = 1 ) |
45 |
9 44
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = ( 𝐶 · 1 ) ) |
46 |
11
|
mulid1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐶 · 1 ) = 𝐶 ) |
47 |
45 46
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = 𝐶 ) |
48 |
43
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 → ¬ 𝑘 ∈ 𝐵 ) ) |
49 |
48
|
con2d |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐵 → ¬ 𝑘 ∈ 𝐴 ) ) |
50 |
49
|
imp |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ¬ 𝑘 ∈ 𝐴 ) |
51 |
50
|
iffalsed |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = 1 ) |
52 |
51 22
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = ( 1 · 𝐶 ) ) |
53 |
24
|
mulid2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( 1 · 𝐶 ) = 𝐶 ) |
54 |
52 53
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐵 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = 𝐶 ) |
55 |
47 54
|
jaodan |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ 𝐴 ∨ 𝑘 ∈ 𝐵 ) ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = 𝐶 ) |
56 |
41 55
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑈 ) → ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = 𝐶 ) |
57 |
56
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑈 ( if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) · if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) = ∏ 𝑘 ∈ 𝑈 𝐶 ) |
58 |
31 37 57
|
3eqtr2rd |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝑈 𝐶 = ( ∏ 𝑘 ∈ 𝐴 𝐶 · ∏ 𝑘 ∈ 𝐵 𝐶 ) ) |