Step |
Hyp |
Ref |
Expression |
1 |
|
fsumshftd.1 |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
2 |
|
fsumshftd.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
fsumshftd.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
4 |
|
fsumshftd.4 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
5 |
|
fsumshftd.5 |
⊢ ( ( 𝜑 ∧ 𝑗 = ( 𝑘 − 𝐾 ) ) → 𝐴 = 𝐵 ) |
6 |
|
nfcv |
⊢ Ⅎ 𝑤 𝐴 |
7 |
|
nfcsb1v |
⊢ Ⅎ 𝑗 ⦋ 𝑤 / 𝑗 ⦌ 𝐴 |
8 |
|
csbeq1a |
⊢ ( 𝑗 = 𝑤 → 𝐴 = ⦋ 𝑤 / 𝑗 ⦌ 𝐴 ) |
9 |
6 7 8
|
cbvsumi |
⊢ Σ 𝑗 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = Σ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ⦋ 𝑤 / 𝑗 ⦌ 𝐴 |
10 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ) |
11 |
7
|
nfel1 |
⊢ Ⅎ 𝑗 ⦋ 𝑤 / 𝑗 ⦌ 𝐴 ∈ ℂ |
12 |
10 11
|
nfim |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ) → ⦋ 𝑤 / 𝑗 ⦌ 𝐴 ∈ ℂ ) |
13 |
|
eleq1w |
⊢ ( 𝑗 = 𝑤 → ( 𝑗 ∈ ( 𝑀 ... 𝑁 ) ↔ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ) ) |
14 |
13
|
anbi2d |
⊢ ( 𝑗 = 𝑤 → ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) ↔ ( 𝜑 ∧ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ) ) ) |
15 |
8
|
eleq1d |
⊢ ( 𝑗 = 𝑤 → ( 𝐴 ∈ ℂ ↔ ⦋ 𝑤 / 𝑗 ⦌ 𝐴 ∈ ℂ ) ) |
16 |
14 15
|
imbi12d |
⊢ ( 𝑗 = 𝑤 → ( ( ( 𝜑 ∧ 𝑗 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ) → ⦋ 𝑤 / 𝑗 ⦌ 𝐴 ∈ ℂ ) ) ) |
17 |
12 16 4
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ) → ⦋ 𝑤 / 𝑗 ⦌ 𝐴 ∈ ℂ ) |
18 |
|
csbeq1 |
⊢ ( 𝑤 = ( 𝑘 − 𝐾 ) → ⦋ 𝑤 / 𝑗 ⦌ 𝐴 = ⦋ ( 𝑘 − 𝐾 ) / 𝑗 ⦌ 𝐴 ) |
19 |
1 2 3 17 18
|
fsumshft |
⊢ ( 𝜑 → Σ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ⦋ 𝑤 / 𝑗 ⦌ 𝐴 = Σ 𝑘 ∈ ( ( 𝑀 + 𝐾 ) ... ( 𝑁 + 𝐾 ) ) ⦋ ( 𝑘 − 𝐾 ) / 𝑗 ⦌ 𝐴 ) |
20 |
|
ovexd |
⊢ ( 𝜑 → ( 𝑘 − 𝐾 ) ∈ V ) |
21 |
20 5
|
csbied |
⊢ ( 𝜑 → ⦋ ( 𝑘 − 𝐾 ) / 𝑗 ⦌ 𝐴 = 𝐵 ) |
22 |
21
|
sumeq2sdv |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( 𝑀 + 𝐾 ) ... ( 𝑁 + 𝐾 ) ) ⦋ ( 𝑘 − 𝐾 ) / 𝑗 ⦌ 𝐴 = Σ 𝑘 ∈ ( ( 𝑀 + 𝐾 ) ... ( 𝑁 + 𝐾 ) ) 𝐵 ) |
23 |
19 22
|
eqtrd |
⊢ ( 𝜑 → Σ 𝑤 ∈ ( 𝑀 ... 𝑁 ) ⦋ 𝑤 / 𝑗 ⦌ 𝐴 = Σ 𝑘 ∈ ( ( 𝑀 + 𝐾 ) ... ( 𝑁 + 𝐾 ) ) 𝐵 ) |
24 |
9 23
|
syl5eq |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = Σ 𝑘 ∈ ( ( 𝑀 + 𝐾 ) ... ( 𝑁 + 𝐾 ) ) 𝐵 ) |