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Mirrors > Home > ILE Home > Th. List > ennnfonelemjn | GIF version |
Description: Lemma for ennnfone 12428. 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 9565 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 0p1e1 9035 | . . . . 5 ⊢ (0 + 1) = 1 | |
3 | 2 | fveq2i 5520 | . . . 4 ⊢ (ℤ≥‘(0 + 1)) = (ℤ≥‘1) |
4 | 1, 3 | eqtr4i 2201 | . . 3 ⊢ ℕ = (ℤ≥‘(0 + 1)) |
5 | 4 | eleq2i 2244 | . 2 ⊢ (𝑓 ∈ ℕ ↔ 𝑓 ∈ (ℤ≥‘(0 + 1))) |
6 | ennnfonelemh.j | . . . 4 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
7 | eqeq1 2184 | . . . . 5 ⊢ (𝑥 = 𝑓 → (𝑥 = 0 ↔ 𝑓 = 0)) | |
8 | fvoveq1 5900 | . . . . 5 ⊢ (𝑥 = 𝑓 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(𝑓 − 1))) | |
9 | 7, 8 | ifbieq2d 3560 | . . . 4 ⊢ (𝑥 = 𝑓 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
10 | nnnn0 9185 | . . . . 5 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ ℕ0) | |
11 | 10 | adantl 277 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑓 ∈ ℕ0) |
12 | nnne0 8949 | . . . . . . . 8 ⊢ (𝑓 ∈ ℕ → 𝑓 ≠ 0) | |
13 | 12 | neneqd 2368 | . . . . . . 7 ⊢ (𝑓 ∈ ℕ → ¬ 𝑓 = 0) |
14 | 13 | iffalsed 3546 | . . . . . 6 ⊢ (𝑓 ∈ ℕ → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) = (◡𝑁‘(𝑓 − 1))) |
15 | 14 | adantl 277 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) = (◡𝑁‘(𝑓 − 1))) |
16 | 0zd 9267 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 0 ∈ ℤ) | |
17 | ennnfonelemh.n | . . . . . . . 8 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
18 | 16, 17 | frec2uzf1od 10408 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑁:ω–1-1-onto→(ℤ≥‘0)) |
19 | f1ocnv 5476 | . . . . . . 7 ⊢ (𝑁:ω–1-1-onto→(ℤ≥‘0) → ◡𝑁:(ℤ≥‘0)–1-1-onto→ω) | |
20 | f1of 5463 | . . . . . . 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 9266 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
23 | 5 | biimpi 120 | . . . . . . . 8 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ (ℤ≥‘(0 + 1))) |
24 | 23 | adantl 277 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑓 ∈ (ℤ≥‘(0 + 1))) |
25 | eluzp1m1 9553 | . . . . . . 7 ⊢ ((0 ∈ ℤ ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝑓 − 1) ∈ (ℤ≥‘0)) | |
26 | 22, 24, 25 | sylancr 414 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝑓 − 1) ∈ (ℤ≥‘0)) |
27 | 21, 26 | ffvelcdmd 5654 | . . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (◡𝑁‘(𝑓 − 1)) ∈ ω) |
28 | 15, 27 | eqeltrd 2254 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1))) ∈ ω) |
29 | 6, 9, 11, 28 | fvmptd3 5611 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝐽‘𝑓) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
30 | 29, 28 | eqeltrd 2254 | . 2 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → (𝐽‘𝑓) ∈ ω) |
31 | 5, 30 | sylan2br 288 | 1 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ∃wrex 2456 ∪ cun 3129 ∅c0 3424 ifcif 3536 {csn 3594 ⟨cop 3597 ↦ cmpt 4066 suc csuc 4367 ωcom 4591 ◡ccnv 4627 dom cdm 4628 “ cima 4631 ⟶wf 5214 –onto→wfo 5216 –1-1-onto→wf1o 5217 ‘cfv 5218 (class class class)co 5877 ∈ cmpo 5879 freccfrec 6393 ↑pm cpm 6651 0cc0 7813 1c1 7814 + caddc 7816 − cmin 8130 ℕcn 8921 ℕ0cn0 9178 ℤcz 9255 ℤ≥cuz 9530 seqcseq 10447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-inn 8922 df-n0 9179 df-z 9256 df-uz 9531 |
This theorem is referenced by: ennnfonelemh 12407 ennnfonelem0 12408 ennnfonelemp1 12409 ennnfonelemom 12411 |
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