Step | Hyp | Ref
| Expression |
1 | | oveq1 5884 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑧 → (𝑐 + 1) = (𝑧 + 1)) |
2 | 1 | fveq2d 5521 |
. . . . . . . . 9
⊢ (𝑐 = 𝑧 → (𝐹‘(𝑐 + 1)) = (𝐹‘(𝑧 + 1))) |
3 | 2 | oveq2d 5893 |
. . . . . . . 8
⊢ (𝑐 = 𝑧 → (𝑑 + (𝐹‘(𝑐 + 1))) = (𝑑 + (𝐹‘(𝑧 + 1)))) |
4 | | oveq1 5884 |
. . . . . . . 8
⊢ (𝑑 = 𝑤 → (𝑑 + (𝐹‘(𝑧 + 1))) = (𝑤 + (𝐹‘(𝑧 + 1)))) |
5 | 3, 4 | cbvmpov 5957 |
. . . . . . 7
⊢ (𝑐 ∈
(ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1)))) = (𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1)))) |
6 | 5 | oveqi 5890 |
. . . . . 6
⊢ (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) |
7 | 6 | opeq2i 3784 |
. . . . 5
⊢
⟨(𝑥 + 1),
(𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩ |
8 | 7 | a1i 9 |
. . . 4
⊢ ((𝑥 ∈
(ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑇) → ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) |
9 | 8 | mpoeq3ia 5942 |
. . 3
⊢ (𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) |
10 | | oveq1 5884 |
. . . . 5
⊢ (𝑥 = 𝑎 → (𝑥 + 1) = (𝑎 + 1)) |
11 | | oveq1 5884 |
. . . . 5
⊢ (𝑥 = 𝑎 → (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)) |
12 | 10, 11 | opeq12d 3788 |
. . . 4
⊢ (𝑥 = 𝑎 → ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) |
13 | | oveq2 5885 |
. . . . 5
⊢ (𝑦 = 𝑏 → (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦) = (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)) |
14 | 13 | opeq2d 3787 |
. . . 4
⊢ (𝑦 = 𝑏 → ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩ = ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩) |
15 | 12, 14 | cbvmpov 5957 |
. . 3
⊢ (𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩) |
16 | 9, 15 | eqtr3i 2200 |
. 2
⊢ (𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩) |
17 | | freceq1 6395 |
. 2
⊢ ((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩) = (𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹‘𝑀)⟩)) |
18 | 16, 17 | ax-mp 5 |
1
⊢
frec((𝑥 ∈
(ℤ≥‘𝑀), 𝑦 ∈ 𝑇 ↦ ⟨(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) = frec((𝑎 ∈ (ℤ≥‘𝑀), 𝑏 ∈ 𝑇 ↦ ⟨(𝑎 + 1), (𝑎(𝑐 ∈ (ℤ≥‘𝑀), 𝑑 ∈ 𝑆 ↦ (𝑑 + (𝐹‘(𝑐 + 1))))𝑏)⟩), ⟨𝑀, (𝐹‘𝑀)⟩) |