![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5899 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 + 1) = (𝑥 + 1)) | |
7 | 6 | cbvmptv 4114 | . . 3 ⊢ (𝑧 ∈ ℤ ↦ (𝑧 + 1)) = (𝑥 ∈ ℤ ↦ (𝑥 + 1)) |
8 | freceq1 6412 | . . 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 10443 | 1 ⊢ ((𝜑 ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝑃‘(𝐵 + 1)) = (𝐵𝐹(𝑃‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ⊆ wss 3144 ⟨cop 3610 ↦ cmpt 4079 ran crn 4642 ‘cfv 5232 (class class class)co 5892 ∈ cmpo 5894 freccfrec 6410 1c1 7832 + caddc 7834 ℤcz 9273 ℤ≥cuz 9548 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7922 ax-resscn 7923 ax-1cn 7924 ax-1re 7925 ax-icn 7926 ax-addcl 7927 ax-addrcl 7928 ax-mulcl 7929 ax-addcom 7931 ax-addass 7933 ax-distr 7935 ax-i2m1 7936 ax-0lt1 7937 ax-0id 7939 ax-rnegex 7940 ax-cnre 7942 ax-pre-ltirr 7943 ax-pre-ltwlin 7944 ax-pre-lttrn 7945 ax-pre-ltadd 7947 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-pnf 8014 df-mnf 8015 df-xr 8016 df-ltxr 8017 df-le 8018 df-sub 8150 df-neg 8151 df-inn 8940 df-n0 9197 df-z 9274 df-uz 9549 |
This theorem is referenced by: seq3p1 10482 seqp1cd 10486 |
Copyright terms: Public domain | W3C validator |