Step |
Hyp |
Ref |
Expression |
1 |
|
hash2iun.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
2 |
|
hash2iun.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ Fin ) |
3 |
|
hash2iun.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝐶 ∈ Fin ) |
4 |
|
hash2iun.da |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ) |
5 |
|
hash2iun.db |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → Disj 𝑦 ∈ 𝐵 𝐶 ) |
6 |
3
|
3expa |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐵 ) → 𝐶 ∈ Fin ) |
7 |
6
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑦 ∈ 𝐵 𝐶 ∈ Fin ) |
8 |
|
iunfi |
⊢ ( ( 𝐵 ∈ Fin ∧ ∀ 𝑦 ∈ 𝐵 𝐶 ∈ Fin ) → ∪ 𝑦 ∈ 𝐵 𝐶 ∈ Fin ) |
9 |
2 7 8
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∪ 𝑦 ∈ 𝐵 𝐶 ∈ Fin ) |
10 |
1 9 4
|
hashiun |
⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ) = Σ 𝑥 ∈ 𝐴 ( ♯ ‘ ∪ 𝑦 ∈ 𝐵 𝐶 ) ) |
11 |
2 6 5
|
hashiun |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ♯ ‘ ∪ 𝑦 ∈ 𝐵 𝐶 ) = Σ 𝑦 ∈ 𝐵 ( ♯ ‘ 𝐶 ) ) |
12 |
11
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑥 ∈ 𝐴 ( ♯ ‘ ∪ 𝑦 ∈ 𝐵 𝐶 ) = Σ 𝑥 ∈ 𝐴 Σ 𝑦 ∈ 𝐵 ( ♯ ‘ 𝐶 ) ) |
13 |
10 12
|
eqtrd |
⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 ) = Σ 𝑥 ∈ 𝐴 Σ 𝑦 ∈ 𝐵 ( ♯ ‘ 𝐶 ) ) |