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Mirrors > Home > ILE Home > Th. List > frec2uzuzd | GIF version |
Description: The value 𝐺 (see frec2uz0d 10355) 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 2239 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ω ↔ 𝐴 ∈ ω)) |
4 | 2 | fveq2d 5500 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐺‘𝑦) = (𝐺‘𝐴)) |
5 | 4 | eleq1d 2239 | . . . 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 5496 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐺‘𝑦) = (𝐺‘∅)) | |
8 | 7 | eleq1d 2239 | . . . . 5 ⊢ (𝑦 = ∅ → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘∅) ∈ (ℤ≥‘𝐶))) |
9 | fveq2 5496 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧)) | |
10 | 9 | eleq1d 2239 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘𝑧) ∈ (ℤ≥‘𝐶))) |
11 | fveq2 5496 | . . . . . 6 ⊢ (𝑦 = suc 𝑧 → (𝐺‘𝑦) = (𝐺‘suc 𝑧)) | |
12 | 11 | eleq1d 2239 | . . . . 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 10355 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | uzid 9501 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶)) | |
17 | 13, 16 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶)) |
18 | 15, 17 | eqeltrd 2247 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ (ℤ≥‘𝐶)) |
19 | peano2uz 9542 | . . . . . . 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 10357 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
23 | 22 | eleq1d 2239 | . . . . . . 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 4585 | . . . 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 2782 | . 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 1348 ∈ wcel 2141 ∅c0 3414 ↦ cmpt 4050 suc csuc 4350 ωcom 4574 ‘cfv 5198 (class class class)co 5853 freccfrec 6369 1c1 7775 + caddc 7777 ℤcz 9212 ℤ≥cuz 9487 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 |
This theorem is referenced by: frec2uzltd 10359 frec2uzrand 10361 frec2uzrdg 10365 frecuzrdgsuc 10370 hashcl 10715 ennnfonelemrn 12374 |
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