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Mirrors > Home > ILE Home > Th. List > frec2uzuzd | GIF version |
Description: The value 𝐺 (see frec2uz0d 10165) 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 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴) | |
3 | 2 | eleq1d 2206 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ω ↔ 𝐴 ∈ ω)) |
4 | 2 | fveq2d 5418 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐺‘𝑦) = (𝐺‘𝐴)) |
5 | 4 | eleq1d 2206 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘𝐴) ∈ (ℤ≥‘𝐶))) |
6 | 3, 5 | imbi12d 233 | . . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ((𝑦 ∈ ω → (𝐺‘𝑦) ∈ (ℤ≥‘𝐶)) ↔ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ (ℤ≥‘𝐶)))) |
7 | fveq2 5414 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐺‘𝑦) = (𝐺‘∅)) | |
8 | 7 | eleq1d 2206 | . . . . 5 ⊢ (𝑦 = ∅ → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘∅) ∈ (ℤ≥‘𝐶))) |
9 | fveq2 5414 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧)) | |
10 | 9 | eleq1d 2206 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘𝑧) ∈ (ℤ≥‘𝐶))) |
11 | fveq2 5414 | . . . . . 6 ⊢ (𝑦 = suc 𝑧 → (𝐺‘𝑦) = (𝐺‘suc 𝑧)) | |
12 | 11 | eleq1d 2206 | . . . . 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 10165 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | uzid 9333 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶)) | |
17 | 13, 16 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶)) |
18 | 15, 17 | eqeltrd 2214 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ (ℤ≥‘𝐶)) |
19 | peano2uz 9371 | . . . . . . 7 ⊢ ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → ((𝐺‘𝑧) + 1) ∈ (ℤ≥‘𝐶)) | |
20 | 13 | adantl 275 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → 𝐶 ∈ ℤ) |
21 | simpl 108 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → 𝑧 ∈ ω) | |
22 | 20, 14, 21 | frec2uzsucd 10167 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
23 | 22 | eleq1d 2206 | . . . . . . 7 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → ((𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶) ↔ ((𝐺‘𝑧) + 1) ∈ (ℤ≥‘𝐶))) |
24 | 19, 23 | syl5ibr 155 | . . . . . 6 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶))) |
25 | 24 | ex 114 | . . . . 5 ⊢ (𝑧 ∈ ω → (𝜑 → ((𝐺‘𝑧) ∈ (ℤ≥‘𝐶) → (𝐺‘suc 𝑧) ∈ (ℤ≥‘𝐶)))) |
26 | 8, 10, 12, 18, 25 | finds2 4510 | . . . 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 2733 | . 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 103 = wceq 1331 ∈ wcel 1480 ∅c0 3358 ↦ cmpt 3984 suc csuc 4282 ωcom 4499 ‘cfv 5118 (class class class)co 5767 freccfrec 6280 1c1 7614 + caddc 7616 ℤcz 9047 ℤ≥cuz 9319 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 |
This theorem is referenced by: frec2uzltd 10169 frec2uzrand 10171 frec2uzrdg 10175 frecuzrdgsuc 10180 hashcl 10520 ennnfonelemrn 11921 |
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