Step |
Hyp |
Ref |
Expression |
1 |
|
esumrnmpt.0 |
⊢ Ⅎ 𝑘 𝐴 |
2 |
|
esumrnmpt.1 |
⊢ ( 𝑦 = 𝐵 → 𝐶 = 𝐷 ) |
3 |
|
esumrnmpt.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
esumrnmpt.3 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐷 ∈ ( 0 [,] +∞ ) ) |
5 |
|
esumrnmpt.4 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ( 𝑊 ∖ { ∅ } ) ) |
6 |
|
esumrnmpt.5 |
⊢ ( 𝜑 → Disj 𝑘 ∈ 𝐴 𝐵 ) |
7 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) |
8 |
7
|
rnmpt |
⊢ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = { 𝑧 ∣ ∃ 𝑘 ∈ 𝐴 𝑧 = 𝐵 } |
9 |
|
esumeq1 |
⊢ ( ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = { 𝑧 ∣ ∃ 𝑘 ∈ 𝐴 𝑧 = 𝐵 } → Σ* 𝑦 ∈ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) 𝐶 = Σ* 𝑦 ∈ { 𝑧 ∣ ∃ 𝑘 ∈ 𝐴 𝑧 = 𝐵 } 𝐶 ) |
10 |
8 9
|
ax-mp |
⊢ Σ* 𝑦 ∈ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) 𝐶 = Σ* 𝑦 ∈ { 𝑧 ∣ ∃ 𝑘 ∈ 𝐴 𝑧 = 𝐵 } 𝐶 |
11 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐶 |
12 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
13 |
12 1 5 6
|
disjdsct |
⊢ ( 𝜑 → Fun ◡ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ) |
14 |
11 12 1 2 3 13 4 5
|
esumc |
⊢ ( 𝜑 → Σ* 𝑘 ∈ 𝐴 𝐷 = Σ* 𝑦 ∈ { 𝑧 ∣ ∃ 𝑘 ∈ 𝐴 𝑧 = 𝐵 } 𝐶 ) |
15 |
10 14
|
eqtr4id |
⊢ ( 𝜑 → Σ* 𝑦 ∈ ran ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) 𝐶 = Σ* 𝑘 ∈ 𝐴 𝐷 ) |