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Mirrors > Home > ILE Home > Th. List > frec2uzuzd | GIF version |
Description: The value 𝐺 (see frec2uz0d 9709) 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 2153 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝑦 ∈ ω ↔ 𝐴 ∈ ω)) |
4 | 2 | fveq2d 5260 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → (𝐺‘𝑦) = (𝐺‘𝐴)) |
5 | 4 | eleq1d 2153 | . . . 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 5256 | . . . . . 6 ⊢ (𝑦 = ∅ → (𝐺‘𝑦) = (𝐺‘∅)) | |
8 | 7 | eleq1d 2153 | . . . . 5 ⊢ (𝑦 = ∅ → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘∅) ∈ (ℤ≥‘𝐶))) |
9 | fveq2 5256 | . . . . . 6 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧)) | |
10 | 9 | eleq1d 2153 | . . . . 5 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘𝑧) ∈ (ℤ≥‘𝐶))) |
11 | fveq2 5256 | . . . . . 6 ⊢ (𝑦 = suc 𝑧 → (𝐺‘𝑦) = (𝐺‘suc 𝑧)) | |
12 | 11 | eleq1d 2153 | . . . . 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 9709 | . . . . . 6 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
16 | uzid 8942 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶)) | |
17 | 13, 16 | syl 14 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (ℤ≥‘𝐶)) |
18 | 15, 17 | eqeltrd 2161 | . . . . 5 ⊢ (𝜑 → (𝐺‘∅) ∈ (ℤ≥‘𝐶)) |
19 | peano2uz 8980 | . . . . . . 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 9711 | . . . . . . . 8 ⊢ ((𝑧 ∈ ω ∧ 𝜑) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
23 | 22 | eleq1d 2153 | . . . . . . 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 4382 | . . . 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 2664 | . 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 1287 ∈ wcel 1436 ∅c0 3272 ↦ cmpt 3868 suc csuc 4159 ωcom 4371 ‘cfv 4972 (class class class)co 5594 freccfrec 6090 1c1 7272 + caddc 7274 ℤcz 8660 ℤ≥cuz 8928 |
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 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-addcom 7366 ax-addass 7368 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-0id 7374 ax-rnegex 7375 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-ltadd 7382 |
This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-iord 4160 df-on 4162 df-ilim 4163 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-recs 6005 df-frec 6091 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-inn 8335 df-n0 8584 df-z 8661 df-uz 8929 |
This theorem is referenced by: frec2uzltd 9713 frec2uzrand 9715 frec2uzrdg 9719 frecuzrdgsuc 9724 hashcl 10038 |
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