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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 982 | . . . . . . 7 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → + = 𝑄) | |
2 | 1 | oveqd 5799 | . . . . . 6 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → (𝑦 + (𝐹‘(𝑥 + 1))) = (𝑦𝑄(𝐹‘(𝑥 + 1)))) |
3 | 2 | opeq2d 3720 | . . . . 5 ⊢ (( + = 𝑄 ∧ 𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ V) → 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉 = 〈(𝑥 + 1), (𝑦𝑄(𝐹‘(𝑥 + 1)))〉) |
4 | 3 | mpoeq3dva 5843 | . . . 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 6297 | . . . 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 4776 | . 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 10250 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑦 + (𝐹‘(𝑥 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
9 | df-seqfrec 10250 | . 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 2198 | 1 ⊢ ( + = 𝑄 → seq𝑀( + , 𝐹) = seq𝑀(𝑄, 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 Vcvv 2689 〈cop 3535 ran crn 4548 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 freccfrec 6295 1c1 7645 + caddc 7647 ℤ≥cuz 9350 seqcseq 10249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-iota 5096 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-recs 6210 df-frec 6296 df-seqfrec 10250 |
This theorem is referenced by: seqeq2d 10256 resqrex 10830 |
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