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Mirrors > Home > ILE Home > Th. List > frec2uzuzd | GIF version |
Description: The value 𝐺 (see frec2uz0d 8866) 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 103 | . . . . 5 ⊢ ((φ ∧ y = A) → y = A) | |
3 | 2 | eleq1d 2103 | . . . 4 ⊢ ((φ ∧ y = A) → (y ∈ 𝜔 ↔ A ∈ 𝜔)) |
4 | 2 | fveq2d 5125 | . . . . 5 ⊢ ((φ ∧ y = A) → (𝐺‘y) = (𝐺‘A)) |
5 | 4 | eleq1d 2103 | . . . 4 ⊢ ((φ ∧ y = A) → ((𝐺‘y) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘A) ∈ (ℤ≥‘𝐶))) |
6 | 3, 5 | imbi12d 223 | . . 3 ⊢ ((φ ∧ y = A) → ((y ∈ 𝜔 → (𝐺‘y) ∈ (ℤ≥‘𝐶)) ↔ (A ∈ 𝜔 → (𝐺‘A) ∈ (ℤ≥‘𝐶)))) |
7 | fveq2 5121 | . . . . . 6 ⊢ (y = ∅ → (𝐺‘y) = (𝐺‘∅)) | |
8 | 7 | eleq1d 2103 | . . . . 5 ⊢ (y = ∅ → ((𝐺‘y) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘∅) ∈ (ℤ≥‘𝐶))) |
9 | fveq2 5121 | . . . . . 6 ⊢ (y = z → (𝐺‘y) = (𝐺‘z)) | |
10 | 9 | eleq1d 2103 | . . . . 5 ⊢ (y = z → ((𝐺‘y) ∈ (ℤ≥‘𝐶) ↔ (𝐺‘z) ∈ (ℤ≥‘𝐶))) |
11 | fveq2 5121 | . . . . . 6 ⊢ (y = suc z → (𝐺‘y) = (𝐺‘suc z)) | |
12 | 11 | eleq1d 2103 | . . . . 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 8866 | . . . . . 6 ⊢ (φ → (𝐺‘∅) = 𝐶) |
16 | uzid 8263 | . . . . . . 7 ⊢ (𝐶 ∈ ℤ → 𝐶 ∈ (ℤ≥‘𝐶)) | |
17 | 13, 16 | syl 14 | . . . . . 6 ⊢ (φ → 𝐶 ∈ (ℤ≥‘𝐶)) |
18 | 15, 17 | eqeltrd 2111 | . . . . 5 ⊢ (φ → (𝐺‘∅) ∈ (ℤ≥‘𝐶)) |
19 | peano2uz 8302 | . . . . . . 7 ⊢ ((𝐺‘z) ∈ (ℤ≥‘𝐶) → ((𝐺‘z) + 1) ∈ (ℤ≥‘𝐶)) | |
20 | 13 | adantl 262 | . . . . . . . . 9 ⊢ ((z ∈ 𝜔 ∧ φ) → 𝐶 ∈ ℤ) |
21 | simpl 102 | . . . . . . . . 9 ⊢ ((z ∈ 𝜔 ∧ φ) → z ∈ 𝜔) | |
22 | 20, 14, 21 | frec2uzsucd 8868 | . . . . . . . 8 ⊢ ((z ∈ 𝜔 ∧ φ) → (𝐺‘suc z) = ((𝐺‘z) + 1)) |
23 | 22 | eleq1d 2103 | . . . . . . 7 ⊢ ((z ∈ 𝜔 ∧ φ) → ((𝐺‘suc z) ∈ (ℤ≥‘𝐶) ↔ ((𝐺‘z) + 1) ∈ (ℤ≥‘𝐶))) |
24 | 19, 23 | syl5ibr 145 | . . . . . 6 ⊢ ((z ∈ 𝜔 ∧ φ) → ((𝐺‘z) ∈ (ℤ≥‘𝐶) → (𝐺‘suc z) ∈ (ℤ≥‘𝐶))) |
25 | 24 | ex 108 | . . . . 5 ⊢ (z ∈ 𝜔 → (φ → ((𝐺‘z) ∈ (ℤ≥‘𝐶) → (𝐺‘suc z) ∈ (ℤ≥‘𝐶)))) |
26 | 8, 10, 12, 18, 25 | finds2 4267 | . . . 4 ⊢ (y ∈ 𝜔 → (φ → (𝐺‘y) ∈ (ℤ≥‘𝐶))) |
27 | 26 | com12 27 | . . 3 ⊢ (φ → (y ∈ 𝜔 → (𝐺‘y) ∈ (ℤ≥‘𝐶))) |
28 | 1, 6, 27 | vtocld 2600 | . 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 97 = wceq 1242 ∈ wcel 1390 ∅c0 3218 ↦ cmpt 3809 suc csuc 4068 𝜔com 4256 ‘cfv 4845 (class class class)co 5455 freccfrec 5917 1c1 6712 + caddc 6714 ℤcz 8021 ℤ≥cuz 8249 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-ltadd 6799 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-frec 5918 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-inn 7696 df-n0 7958 df-z 8022 df-uz 8250 |
This theorem is referenced by: frec2uzltd 8870 frec2uzrand 8872 frec2uzrdg 8876 frecuzrdgsuc 8882 |
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