Step |
Hyp |
Ref |
Expression |
1 |
|
fprodm1s.1 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
2 |
|
fprodm1s.2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
3 |
2
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 ∈ ℂ ) |
4 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑚 / 𝑘 ⦌ 𝐴 |
5 |
4
|
nfel1 |
⊢ Ⅎ 𝑘 ⦋ 𝑚 / 𝑘 ⦌ 𝐴 ∈ ℂ |
6 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑚 → 𝐴 = ⦋ 𝑚 / 𝑘 ⦌ 𝐴 ) |
7 |
6
|
eleq1d |
⊢ ( 𝑘 = 𝑚 → ( 𝐴 ∈ ℂ ↔ ⦋ 𝑚 / 𝑘 ⦌ 𝐴 ∈ ℂ ) ) |
8 |
5 7
|
rspc |
⊢ ( 𝑚 ∈ ( 𝑀 ... 𝑁 ) → ( ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 ∈ ℂ → ⦋ 𝑚 / 𝑘 ⦌ 𝐴 ∈ ℂ ) ) |
9 |
3 8
|
mpan9 |
⊢ ( ( 𝜑 ∧ 𝑚 ∈ ( 𝑀 ... 𝑁 ) ) → ⦋ 𝑚 / 𝑘 ⦌ 𝐴 ∈ ℂ ) |
10 |
|
csbeq1 |
⊢ ( 𝑚 = 𝑁 → ⦋ 𝑚 / 𝑘 ⦌ 𝐴 = ⦋ 𝑁 / 𝑘 ⦌ 𝐴 ) |
11 |
1 9 10
|
fprodm1 |
⊢ ( 𝜑 → ∏ 𝑚 ∈ ( 𝑀 ... 𝑁 ) ⦋ 𝑚 / 𝑘 ⦌ 𝐴 = ( ∏ 𝑚 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ⦋ 𝑚 / 𝑘 ⦌ 𝐴 · ⦋ 𝑁 / 𝑘 ⦌ 𝐴 ) ) |
12 |
|
nfcv |
⊢ Ⅎ 𝑚 𝐴 |
13 |
12 4 6
|
cbvprodi |
⊢ ∏ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ∏ 𝑚 ∈ ( 𝑀 ... 𝑁 ) ⦋ 𝑚 / 𝑘 ⦌ 𝐴 |
14 |
12 4 6
|
cbvprodi |
⊢ ∏ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) 𝐴 = ∏ 𝑚 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ⦋ 𝑚 / 𝑘 ⦌ 𝐴 |
15 |
14
|
oveq1i |
⊢ ( ∏ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) 𝐴 · ⦋ 𝑁 / 𝑘 ⦌ 𝐴 ) = ( ∏ 𝑚 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) ⦋ 𝑚 / 𝑘 ⦌ 𝐴 · ⦋ 𝑁 / 𝑘 ⦌ 𝐴 ) |
16 |
11 13 15
|
3eqtr4g |
⊢ ( 𝜑 → ∏ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( ∏ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) 𝐴 · ⦋ 𝑁 / 𝑘 ⦌ 𝐴 ) ) |