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Mirrors > Home > ILE Home > Th. List > frecuzrdgsuct | GIF version |
Description: Successor value of a recursive definition generator on upper integers. (Contributed by Jim Kingdon, 29-Apr-2022.) |
Ref | Expression |
---|---|
frecuzrdgrclt.c | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frecuzrdgrclt.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
frecuzrdgrclt.t | ⊢ (𝜑 → 𝑆 ⊆ 𝑇) |
frecuzrdgrclt.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
frecuzrdgrclt.r | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
frecuzrdgsuct.ran | ⊢ (𝜑 → 𝑃 = ran 𝑅) |
Ref | Expression |
---|---|
frecuzrdgsuct | ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frecuzrdgrclt.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
2 | frecuzrdgrclt.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | frecuzrdgrclt.t | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | |
4 | frecuzrdgrclt.f | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
5 | frecuzrdgrclt.r | . 2 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑇 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
6 | oveq1 5749 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 + 1) = (𝑥 + 1)) | |
7 | 6 | cbvmptv 3994 | . . 3 ⊢ (𝑧 ∈ ℤ ↦ (𝑧 + 1)) = (𝑥 ∈ ℤ ↦ (𝑥 + 1)) |
8 | freceq1 6257 | . . 3 ⊢ ((𝑧 ∈ ℤ ↦ (𝑧 + 1)) = (𝑥 ∈ ℤ ↦ (𝑥 + 1)) → frec((𝑧 ∈ ℤ ↦ (𝑧 + 1)), 𝐶) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ frec((𝑧 ∈ ℤ ↦ (𝑧 + 1)), 𝐶) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
10 | frecuzrdgsuct.ran | . 2 ⊢ (𝜑 → 𝑃 = ran 𝑅) | |
11 | 1, 2, 3, 4, 5, 9, 10 | frecuzrdgsuctlem 10164 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 〈cop 3500 ↦ cmpt 3959 ran crn 4510 ‘cfv 5093 (class class class)co 5742 ∈ cmpo 5744 freccfrec 6255 1c1 7589 + caddc 7591 ℤcz 9022 ℤ≥cuz 9294 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 |
This theorem is referenced by: seq3p1 10203 seqp1cd 10207 |
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