Step |
Hyp |
Ref |
Expression |
1 |
|
fprodcllemf.ph |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
fprodcllemf.s |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
3 |
|
fprodcllemf.xy |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 · 𝑦 ) ∈ 𝑆 ) |
4 |
|
fprodcllemf.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
|
fprodcllemf.b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑆 ) |
6 |
|
fprodcllemf.1 |
⊢ ( 𝜑 → 1 ∈ 𝑆 ) |
7 |
|
nfcv |
⊢ Ⅎ 𝑗 𝐵 |
8 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 |
9 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑗 → 𝐵 = ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ) |
10 |
7 8 9
|
cbvprodi |
⊢ ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 |
11 |
5
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 → 𝐵 ∈ 𝑆 ) ) |
12 |
1 11
|
ralrimi |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ) |
13 |
|
rspsbc |
⊢ ( 𝑗 ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 → [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ) ) |
14 |
12 13
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ) |
15 |
|
sbcel1g |
⊢ ( 𝑗 ∈ V → ( [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 ) ) |
16 |
15
|
elv |
⊢ ( [ 𝑗 / 𝑘 ] 𝐵 ∈ 𝑆 ↔ ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 ) |
17 |
14 16
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 ) |
18 |
2 3 4 17 6
|
fprodcllem |
⊢ ( 𝜑 → ∏ 𝑗 ∈ 𝐴 ⦋ 𝑗 / 𝑘 ⦌ 𝐵 ∈ 𝑆 ) |
19 |
10 18
|
eqeltrid |
⊢ ( 𝜑 → ∏ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆 ) |