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Mirrors > Home > ILE Home > Th. List > frec2uzuzd | GIF version |
Description: The value 𝐺 (see frec2uz0d 9558) at an ordinal natural number is in the upper integers. (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
frec2uzzd.a | ⊢ (𝜑 → 𝐴 ∈ ω) |
Ref | Expression |
---|---|
frec2uzuzd | ⊢ (𝜑 → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uzzd.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ω) | |
2 | simpr 108 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
3 | 2 | eleq1d 2151 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ω ↔ 𝐴 ∈ ω)) |
4 | 2 | fveq2d 5235 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐺‘𝑦) = (𝐺‘𝐴)) |
5 | 4 | eleq1d 2151 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘𝐴) ∈ (ℤ≥‘𝐶))) |
6 | 3, 5 | imbi12d 232 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶)) ↔ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶)))) |
7 | fveq2 5231 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐺‘𝑦) = (𝐺‘∅)) | |
8 | 7 | eleq1d 2151 | . . . . 5 ⊢ (𝑦 = ∅ → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘∅) ∈ (ℤ≥‘𝐶))) |
9 | fveq2 5231 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧)) | |
10 | 9 | eleq1d 2151 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘𝑧) ∈ (ℤ≥‘𝐶))) |
11 | fveq2 5231 | . . . . . 6 ⊢ (𝑦 = suc 𝑧 → (𝐺‘𝑦) = (𝐺‘suc 𝑧)) | |
12 | 11 | eleq1d 2151 | . . . . 5 ⊢ (𝑦 = suc 𝑧 → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶))) |
13 | frec2uz.1 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
14 | frec2uz.2 | . . . . . . 7 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
15 | 13, 14 | frec2uz0d 9558 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | uzid 8791 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶)) | |
17 | 13, 16 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶)) |
18 | 15, 17 | eqeltrd 2159 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ (ℤ≥‘𝐶)) |
19 | peano2uz 8829 | . . . . . . 7 ⊢ ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → ((𝐺‘𝑧) + 1) ∈ (ℤ≥‘𝐶)) | |
20 | 13 | adantl 271 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → 𝐶 ∈ ℤ) |
21 | simpl 107 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → 𝑧 ∈ ω) | |
22 | 20, 14, 21 | frec2uzsucd 9560 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
23 | 22 | eleq1d 2151 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → ((𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶) ↔ ((𝐺‘𝑧) + 1) ∈ (ℤ≥‘𝐶))) |
24 | 19, 23 | syl5ibr 154 | . . . . . 6 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶))) |
25 | 24 | ex 113 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝜑 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶)))) |
26 | 8, 10, 12, 18, 25 | finds2 4371 | . . . 4 ⊢ (𝑦 ∈ ω → (𝜑 → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶))) |
27 | 26 | com12 30 | . . 3 ⊢ (𝜑 → (𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶))) |
28 | 1, 6, 27 | vtocld 2661 | . 2 ⊢ (𝜑 → (𝐴 ∈ ω → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶))) |
29 | 1, 28 | mpd 13 | 1 ⊢ (𝜑 → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 ∅c0 3268 ↦ cmpt 3860 suc csuc 4149 ωcom 4360 ‘cfv 4953 (class class class)co 5565 freccfrec 6061 1c1 7121 + caddc 7123 ℤcz 8509 ℤ≥cuz 8777 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 ax-cnex 7206 ax-resscn 7207 ax-1cn 7208 ax-1re 7209 ax-icn 7210 ax-addcl 7211 ax-addrcl 7212 ax-mulcl 7213 ax-addcom 7215 ax-addass 7217 ax-distr 7219 ax-i2m1 7220 ax-0lt1 7221 ax-0id 7223 ax-rnegex 7224 ax-cnre 7226 ax-pre-ltirr 7227 ax-pre-ltwlin 7228 ax-pre-lttrn 7229 ax-pre-ltadd 7231 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-id 4077 df-iord 4150 df-on 4152 df-ilim 4153 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-riota 5521 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-recs 5976 df-frec 6062 df-pnf 7294 df-mnf 7295 df-xr 7296 df-ltxr 7297 df-le 7298 df-sub 7425 df-neg 7426 df-inn 8184 df-n0 8433 df-z 8510 df-uz 8778 |
This theorem is referenced by: frec2uzltd 9562 frec2uzrand 9564 frec2uzrdg 9568 frecuzrdgsuc 9573 hashcl 9882 |
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