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Mirrors > Home > ILE Home > Th. List > ennnfonelemjn | GIF version |
Description: Lemma for ennnfone 11938. Non-initial state for 𝐽. (Contributed by Jim Kingdon, 20-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemh.dceq | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
ennnfonelemh.f | ⊢ (𝜑 → 𝐹:ω–onto→𝐴) |
ennnfonelemh.ne | ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) |
ennnfonelemh.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {〈dom 𝑥, (𝐹‘𝑦)〉}))) |
ennnfonelemh.n | ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
ennnfonelemh.j | ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) |
ennnfonelemh.h | ⊢ 𝐻 = seq0(𝐺, 𝐽) |
Ref | Expression |
---|---|
ennnfonelemjn | ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9361 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 0p1e1 8834 | . . . . 5 ⊢ (0 + 1) = 1 | |
3 | 2 | fveq2i 5424 | . . . 4 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
4 | 1, 3 | eqtr4i 2163 | . . 3 ⊢ ℕ = (ℤ≥‘(0 + 1)) |
5 | 4 | eleq2i 2206 | . 2 ⊢ (𝑓 ∈ ℕ ↔ 𝑓 ∈ (ℤ≥‘(0 + 1))) |
6 | ennnfonelemh.j | . . . 4 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
7 | eqeq1 2146 | . . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = 0 ↔ 𝑓 = 0)) | |
8 | fvoveq1 5797 | . . . . 5 ⊢ (𝑥 = 𝑓 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(𝑓 − 1))) | |
9 | 7, 8 | ifbieq2d 3496 | . . . 4 ⊢ (𝑥 = 𝑓 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
10 | nnnn0 8984 | . . . . 5 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ ℕ0) | |
11 | 10 | adantl 275 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑓 ∈ ℕ0) |
12 | nnne0 8748 | . . . . . . . 8 ⊢ (𝑓 ∈ ℕ → 𝑓 ≠ 0) | |
13 | 12 | neneqd 2329 | . . . . . . 7 ⊢ (𝑓 ∈ ℕ → ¬ 𝑓 = 0) |
14 | 13 | iffalsed 3484 | . . . . . 6 ⊢ (𝑓 ∈ ℕ → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) = (◡𝑁‘(𝑓 − 1))) |
15 | 14 | adantl 275 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) = (◡𝑁‘(𝑓 − 1))) |
16 | 0zd 9066 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 0 ∈ ℤ) | |
17 | ennnfonelemh.n | . . . . . . . 8 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
18 | 16, 17 | frec2uzf1od 10179 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑁:ω–1-1-onto→(ℤ≥‘0)) |
19 | f1ocnv 5380 | . . . . . . 7 ⊢ (𝑁:ω–1-1-onto→(ℤ≥‘0) → ◡𝑁:(ℤ≥‘0)–1-1-onto→ω) | |
20 | f1of 5367 | . . . . . . 7 ⊢ (◡𝑁:(ℤ≥‘0)–1-1-onto→ω → ◡𝑁:(ℤ≥‘0)⟶ω) | |
21 | 18, 19, 20 | 3syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → ◡𝑁:(ℤ≥‘0)⟶ω) |
22 | 0z 9065 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
23 | 5 | biimpi 119 | . . . . . . . 8 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ (ℤ≥‘(0 + 1))) |
24 | 23 | adantl 275 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑓 ∈ (ℤ≥‘(0 + 1))) |
25 | eluzp1m1 9349 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝑓 − 1) ∈ (ℤ≥‘0)) | |
26 | 22, 24, 25 | sylancr 410 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝑓 − 1) ∈ (ℤ≥‘0)) |
27 | 21, 26 | ffvelrnd 5556 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (◡𝑁‘(𝑓 − 1)) ∈ ω) |
28 | 15, 27 | eqeltrd 2216 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) ∈ ω) |
29 | 6, 9, 11, 28 | fvmptd3 5514 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝐽‘𝑓) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
30 | 29, 28 | eqeltrd 2216 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝐽‘𝑓) ∈ ω) |
31 | 5, 30 | sylan2br 286 | 1 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 819 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∀wral 2416 ∃wrex 2417 ∪ cun 3069 ∅c0 3363 ifcif 3474 {csn 3527 〈cop 3530 ↦ cmpt 3989 suc csuc 4287 ωcom 4504 ◡ccnv 4538 dom cdm 4539 “ cima 4542 ⟶wf 5119 –onto→wfo 5121 –1-1-onto→wf1o 5122 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 freccfrec 6287 ↑pm cpm 6543 0cc0 7620 1c1 7621 + caddc 7623 − cmin 7933 ℕcn 8720 ℕ0cn0 8977 ℤcz 9054 ℤ≥cuz 9326 seqcseq 10218 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 |
This theorem is referenced by: ennnfonelemh 11917 ennnfonelem0 11918 ennnfonelemp1 11919 ennnfonelemom 11921 |
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