Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > frecuzrdgrom | GIF version |
Description: The function 𝑅 (used in the definition of the recursive definition generator on upper integers) is a function defined for all natural numbers. (Contributed by Jim Kingdon, 26-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
uzrdg.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
uzrdg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
uzrdg.f | ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝐶) ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
uzrdg.2 | ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) |
Ref | Expression |
---|---|
frecuzrdgrom | ⊢ (𝜑 → 𝑅 Fn ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 8360 | . . . . . . 7 ⊢ ℤ ∈ V | |
2 | uzssz 8638 | . . . . . . 7 ⊢ (ℤ≥‘𝐶) ⊆ ℤ | |
3 | 1, 2 | ssexi 3916 | . . . . . 6 ⊢ (ℤ≥‘𝐶) ∈ V |
4 | uzrdg.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
5 | mpt2exga 5855 | . . . . . 6 ⊢ (((ℤ≥‘𝐶) ∈ V ∧ 𝑆 ∈ 𝑉) → (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉) ∈ V) | |
6 | 3, 4, 5 | sylancr 405 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉) ∈ V) |
7 | vex 2604 | . . . . 5 ⊢ 𝑧 ∈ V | |
8 | fvexg 5214 | . . . . 5 ⊢ (((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉) ∈ V ∧ 𝑧 ∈ V) → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘𝑧) ∈ V) | |
9 | 6, 7, 8 | sylancl 404 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘𝑧) ∈ V) |
10 | 9 | alrimiv 1795 | . . 3 ⊢ (𝜑 → ∀𝑧((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘𝑧) ∈ V) |
11 | frec2uz.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
12 | uzid 8633 | . . . . 5 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶)) | |
13 | 11, 12 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶)) |
14 | uzrdg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
15 | opelxp 4392 | . . . 4 ⊢ (〈𝐶, 𝐴〉 ∈ ((ℤ≥‘𝐶) × 𝑆) ↔ (𝐶 ∈ (ℤ≥‘𝐶) ∧ 𝐴 ∈ 𝑆)) | |
16 | 13, 14, 15 | sylanbrc 408 | . . 3 ⊢ (𝜑 → 〈𝐶, 𝐴〉 ∈ ((ℤ≥‘𝐶) × 𝑆)) |
17 | frecfnom 6009 | . . 3 ⊢ ((∀𝑧((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘𝑧) ∈ V ∧ 〈𝐶, 𝐴〉 ∈ ((ℤ≥‘𝐶) × 𝑆)) → frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) Fn ω) | |
18 | 10, 16, 17 | syl2anc 403 | . 2 ⊢ (𝜑 → frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) Fn ω) |
19 | uzrdg.2 | . . 3 ⊢ 𝑅 = frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) | |
20 | 19 | fneq1i 5013 | . 2 ⊢ (𝑅 Fn ω ↔ frec((𝑥 ∈ (ℤ≥‘𝐶), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) Fn ω) |
21 | 18, 20 | sylibr 132 | 1 ⊢ (𝜑 → 𝑅 Fn ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∀wal 1282 = wceq 1284 ∈ wcel 1433 Vcvv 2601 〈cop 3401 ↦ cmpt 3839 ωcom 4331 × cxp 4361 Fn wfn 4917 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 freccfrec 6000 1c1 6982 + caddc 6984 ℤcz 8351 ℤ≥cuz 8619 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-neg 7282 df-z 8352 df-uz 8620 |
This theorem is referenced by: frecuzrdglem 9413 frecuzrdgfn 9414 frecuzrdg0 9416 |
Copyright terms: Public domain | W3C validator |