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| Mirrors > Home > ILE Home > Th. List > seqeq2 | GIF version | ||
| Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.) |
| Ref | Expression |
|---|---|
| seqeq2 | ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1000 | . . . . . . 7 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → + = 𝑄) | |
| 2 | 1 | oveqd 5986 | . . . . . 6 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1)))) |
| 3 | 2 | opeq2d 3841 | . . . . 5 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉) |
| 4 | 3 | mpoeq3dva 6034 | . . . 4 ⊢ ( + = 𝑄 → (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉)) |
| 5 | freceq1 6503 | . . . 4 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉) = (𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉) → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) | |
| 6 | 4, 5 | syl 14 | . . 3 ⊢ ( + = 𝑄 → frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) |
| 7 | 6 | rneqd 4927 | . 2 ⊢ ( + = 𝑄 → ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉)) |
| 8 | df-seqfrec 10632 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
| 9 | df-seqfrec 10632 | . 2 ⊢ seq𝑀(𝑄, 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
| 10 | 7, 8, 9 | 3eqtr4g 2265 | 1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ w3a 981 = wceq 1373 ∈ wcel 2178 Vcvv 2777 〈cop 3647 ran crn 4695 ‘cfv 5291 (class class class)co 5969 ∈ cmpo 5971 freccfrec 6501 1c1 7963 + caddc 7965 ℤ≥cuz 9685 seqcseq 10631 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-ext 2189 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ral 2491 df-rex 2492 df-v 2779 df-un 3179 df-in 3181 df-ss 3188 df-sn 3650 df-pr 3651 df-op 3653 df-uni 3866 df-br 4061 df-opab 4123 df-mpt 4124 df-cnv 4702 df-dm 4704 df-rn 4705 df-res 4706 df-iota 5252 df-fv 5299 df-ov 5972 df-oprab 5973 df-mpo 5974 df-recs 6416 df-frec 6502 df-seqfrec 10632 |
| This theorem is referenced by: seqeq2d 10638 resqrex 11498 nninfdc 12985 |
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