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Mirrors > Home > ILE Home > Th. List > ennnfonelemjn | GIF version |
Description: Lemma for ennnfone 11974. 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 9385 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 0p1e1 8858 | . . . . 5 ⊢ (0 + 1) = 1 | |
3 | 2 | fveq2i 5432 | . . . 4 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
4 | 1, 3 | eqtr4i 2164 | . . 3 ⊢ ℕ = (ℤ≥‘(0 + 1)) |
5 | 4 | eleq2i 2207 | . 2 ⊢ (𝑓 ∈ ℕ ↔ 𝑓 ∈ (ℤ≥‘(0 + 1))) |
6 | ennnfonelemh.j | . . . 4 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
7 | eqeq1 2147 | . . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = 0 ↔ 𝑓 = 0)) | |
8 | fvoveq1 5805 | . . . . 5 ⊢ (𝑥 = 𝑓 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(𝑓 − 1))) | |
9 | 7, 8 | ifbieq2d 3501 | . . . 4 ⊢ (𝑥 = 𝑓 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
10 | nnnn0 9008 | . . . . 5 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ ℕ0) | |
11 | 10 | adantl 275 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑓 ∈ ℕ0) |
12 | nnne0 8772 | . . . . . . . 8 ⊢ (𝑓 ∈ ℕ → 𝑓 ≠ 0) | |
13 | 12 | neneqd 2330 | . . . . . . 7 ⊢ (𝑓 ∈ ℕ → ¬ 𝑓 = 0) |
14 | 13 | iffalsed 3489 | . . . . . 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 9090 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 0 ∈ ℤ) | |
17 | ennnfonelemh.n | . . . . . . . 8 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
18 | 16, 17 | frec2uzf1od 10210 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑁:ω–1-1-onto→(ℤ≥‘0)) |
19 | f1ocnv 5388 | . . . . . . 7 ⊢ (𝑁:ω–1-1-onto→(ℤ≥‘0) → ◡𝑁:(ℤ≥‘0)–1-1-onto→ω) | |
20 | f1of 5375 | . . . . . . 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 9089 | . . . . . . 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 9373 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝑓 − 1) ∈ (ℤ≥‘0)) | |
26 | 22, 24, 25 | sylancr 411 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝑓 − 1) ∈ (ℤ≥‘0)) |
27 | 21, 26 | ffvelrnd 5564 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (◡𝑁‘(𝑓 − 1)) ∈ ω) |
28 | 15, 27 | eqeltrd 2217 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) ∈ ω) |
29 | 6, 9, 11, 28 | fvmptd3 5522 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝐽‘𝑓) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
30 | 29, 28 | eqeltrd 2217 | . 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 820 = wceq 1332 ∈ wcel 1481 ≠ wne 2309 ∀wral 2417 ∃wrex 2418 ∪ cun 3074 ∅c0 3368 ifcif 3479 {csn 3532 〈cop 3535 ↦ cmpt 3997 suc csuc 4295 ωcom 4512 ◡ccnv 4546 dom cdm 4547 “ cima 4550 ⟶wf 5127 –onto→wfo 5129 –1-1-onto→wf1o 5130 ‘cfv 5131 (class class class)co 5782 ∈ cmpo 5784 freccfrec 6295 ↑pm cpm 6551 0cc0 7644 1c1 7645 + caddc 7647 − cmin 7957 ℕcn 8744 ℕ0cn0 9001 ℤcz 9078 ℤ≥cuz 9350 seqcseq 10249 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 |
This theorem is referenced by: ennnfonelemh 11953 ennnfonelem0 11954 ennnfonelemp1 11955 ennnfonelemom 11957 |
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