Step |
Hyp |
Ref |
Expression |
1 |
|
fprodsplit1f.kph |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
fprodsplit1f.fk |
⊢ ( 𝜑 → Ⅎ 𝑘 𝐷 ) |
3 |
|
fprodsplit1f.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
4 |
|
fprodsplit1f.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
5 |
|
fprodsplit1f.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
6 |
|
fprodsplit1f.d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝐶 ) → 𝐵 = 𝐷 ) |
7 |
|
disjdif |
⊢ ( { 𝐶 } ∩ ( 𝐴 ∖ { 𝐶 } ) ) = ∅ |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( { 𝐶 } ∩ ( 𝐴 ∖ { 𝐶 } ) ) = ∅ ) |
9 |
5
|
snssd |
⊢ ( 𝜑 → { 𝐶 } ⊆ 𝐴 ) |
10 |
|
undif |
⊢ ( { 𝐶 } ⊆ 𝐴 ↔ ( { 𝐶 } ∪ ( 𝐴 ∖ { 𝐶 } ) ) = 𝐴 ) |
11 |
9 10
|
sylib |
⊢ ( 𝜑 → ( { 𝐶 } ∪ ( 𝐴 ∖ { 𝐶 } ) ) = 𝐴 ) |
12 |
11
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( { 𝐶 } ∪ ( 𝐴 ∖ { 𝐶 } ) ) ) |
13 |
1 8 12 3 4
|
fprodsplitf |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ( ∏ 𝑘 ∈ { 𝐶 } 𝐵 · ∏ 𝑘 ∈ ( 𝐴 ∖ { 𝐶 } ) 𝐵 ) ) |
14 |
5
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) ) |
15 |
|
nfv |
⊢ Ⅎ 𝑘 𝐶 ∈ 𝐴 |
16 |
1 15
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) |
17 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝐶 / 𝑘 ⦌ 𝐵 |
18 |
17
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ |
19 |
16 18
|
nfim |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ ) |
20 |
|
eleq1 |
⊢ ( 𝑘 = 𝐶 → ( 𝑘 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴 ) ) |
21 |
20
|
anbi2d |
⊢ ( 𝑘 = 𝐶 → ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) ) ) |
22 |
|
csbeq1a |
⊢ ( 𝑘 = 𝐶 → 𝐵 = ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ) |
23 |
22
|
eleq1d |
⊢ ( 𝑘 = 𝐶 → ( 𝐵 ∈ ℂ ↔ ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) |
24 |
21 23
|
imbi12d |
⊢ ( 𝑘 = 𝐶 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) ) |
25 |
19 24 4
|
vtoclg1f |
⊢ ( 𝐶 ∈ 𝐴 → ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ ) ) |
26 |
5 14 25
|
sylc |
⊢ ( 𝜑 → ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ ) |
27 |
|
prodsns |
⊢ ( ( 𝐶 ∈ 𝐴 ∧ ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ∈ ℂ ) → ∏ 𝑘 ∈ { 𝐶 } 𝐵 = ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ) |
28 |
5 26 27
|
syl2anc |
⊢ ( 𝜑 → ∏ 𝑘 ∈ { 𝐶 } 𝐵 = ⦋ 𝐶 / 𝑘 ⦌ 𝐵 ) |
29 |
1 2 5 6
|
csbiedf |
⊢ ( 𝜑 → ⦋ 𝐶 / 𝑘 ⦌ 𝐵 = 𝐷 ) |
30 |
28 29
|
eqtrd |
⊢ ( 𝜑 → ∏ 𝑘 ∈ { 𝐶 } 𝐵 = 𝐷 ) |
31 |
30
|
oveq1d |
⊢ ( 𝜑 → ( ∏ 𝑘 ∈ { 𝐶 } 𝐵 · ∏ 𝑘 ∈ ( 𝐴 ∖ { 𝐶 } ) 𝐵 ) = ( 𝐷 · ∏ 𝑘 ∈ ( 𝐴 ∖ { 𝐶 } ) 𝐵 ) ) |
32 |
13 31
|
eqtrd |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 = ( 𝐷 · ∏ 𝑘 ∈ ( 𝐴 ∖ { 𝐶 } ) 𝐵 ) ) |