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Mirrors > Home > ILE Home > Th. List > ennnfonelemjn | GIF version |
Description: Lemma for ennnfone 12417. 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 9558 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 0p1e1 9028 | . . . . 5 ⊢ (0 + 1) = 1 | |
3 | 2 | fveq2i 5516 | . . . 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 5894 | . . . . 5 ⊢ (𝑥 = 𝑓 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(𝑓 − 1))) | |
9 | 7, 8 | ifbieq2d 3558 | . . . 4 ⊢ (𝑥 = 𝑓 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(𝑓 = 0, ∅, (◡𝑁‘(𝑓 − 1)))) |
10 | nnnn0 9178 | . . . . 5 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ ℕ0) | |
11 | 10 | adantl 277 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑓 ∈ ℕ0) |
12 | nnne0 8942 | . . . . . . . 8 ⊢ (𝑓 ∈ ℕ → 𝑓 ≠ 0) | |
13 | 12 | neneqd 2368 | . . . . . . 7 ⊢ (𝑓 ∈ ℕ → ¬ 𝑓 = 0) |
14 | 13 | iffalsed 3544 | . . . . . 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 9260 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 0 ∈ ℤ) | |
17 | ennnfonelemh.n | . . . . . . . 8 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
18 | 16, 17 | frec2uzf1od 10400 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ℕ) → 𝑁:ω–1-1-onto→(ℤ≥‘0)) |
19 | f1ocnv 5472 | . . . . . . 7 ⊢ (𝑁:ω–1-1-onto→(ℤ≥‘0) → ◡𝑁:(ℤ≥‘0)–1-1-onto→ω) | |
20 | f1of 5459 | . . . . . . 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 9259 | . . . . . . 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 9546 | . . . . . . 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 5650 | . . . . 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 5607 | . . 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 3127 ∅c0 3422 ifcif 3534 {csn 3592 〈cop 3595 ↦ cmpt 4063 suc csuc 4364 ωcom 4588 ◡ccnv 4624 dom cdm 4625 “ cima 4628 ⟶wf 5210 –onto→wfo 5212 –1-1-onto→wf1o 5213 ‘cfv 5214 (class class class)co 5871 ∈ cmpo 5873 freccfrec 6387 ↑pm cpm 6645 0cc0 7807 1c1 7808 + caddc 7810 − cmin 8123 ℕcn 8914 ℕ0cn0 9171 ℤcz 9248 ℤ≥cuz 9523 seqcseq 10439 |
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 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-addcom 7907 ax-addass 7909 ax-distr 7911 ax-i2m1 7912 ax-0lt1 7913 ax-0id 7915 ax-rnegex 7916 ax-cnre 7918 ax-pre-ltirr 7919 ax-pre-ltwlin 7920 ax-pre-lttrn 7921 ax-pre-ltadd 7923 |
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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-riota 5827 df-ov 5874 df-oprab 5875 df-mpo 5876 df-recs 6302 df-frec 6388 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992 df-le 7993 df-sub 8125 df-neg 8126 df-inn 8915 df-n0 9172 df-z 9249 df-uz 9524 |
This theorem is referenced by: ennnfonelemh 12396 ennnfonelem0 12397 ennnfonelemp1 12398 ennnfonelemom 12400 |
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