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Mirrors > Home > ILE Home > Th. List > ennnfonelemhom | Unicode version |
Description: Lemma for ennnfone 12301. The sequences in increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemh.dceq | DECID |
ennnfonelemh.f | |
ennnfonelemh.ne | |
ennnfonelemh.g | |
ennnfonelemh.n | frec |
ennnfonelemh.j | |
ennnfonelemh.h | |
ennnfonelemhom.m |
Ref | Expression |
---|---|
ennnfonelemhom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemhom.m | . 2 | |
2 | eleq1 2227 | . . . . 5 | |
3 | 2 | rexbidv 2465 | . . . 4 |
4 | 3 | imbi2d 229 | . . 3 |
5 | eleq1 2227 | . . . . 5 | |
6 | 5 | rexbidv 2465 | . . . 4 |
7 | 6 | imbi2d 229 | . . 3 |
8 | eleq1 2227 | . . . . 5 | |
9 | 8 | rexbidv 2465 | . . . 4 |
10 | 9 | imbi2d 229 | . . 3 |
11 | eleq1 2227 | . . . . 5 | |
12 | 11 | rexbidv 2465 | . . . 4 |
13 | 12 | imbi2d 229 | . . 3 |
14 | 1nn0 9121 | . . . 4 | |
15 | 0ex 4103 | . . . . . 6 | |
16 | 15 | snid 3601 | . . . . 5 |
17 | ennnfonelemh.dceq | . . . . . . . 8 DECID | |
18 | ennnfonelemh.f | . . . . . . . 8 | |
19 | ennnfonelemh.ne | . . . . . . . 8 | |
20 | ennnfonelemh.g | . . . . . . . 8 | |
21 | ennnfonelemh.n | . . . . . . . 8 frec | |
22 | ennnfonelemh.j | . . . . . . . 8 | |
23 | ennnfonelemh.h | . . . . . . . 8 | |
24 | 17, 18, 19, 20, 21, 22, 23 | ennnfonelem1 12283 | . . . . . . 7 |
25 | 24 | dmeqd 4800 | . . . . . 6 |
26 | peano1 4565 | . . . . . . . 8 | |
27 | fof 5404 | . . . . . . . . . 10 | |
28 | 18, 27 | syl 14 | . . . . . . . . 9 |
29 | 26 | a1i 9 | . . . . . . . . 9 |
30 | 28, 29 | ffvelrnd 5615 | . . . . . . . 8 |
31 | fnsng 5229 | . . . . . . . 8 | |
32 | 26, 30, 31 | sylancr 411 | . . . . . . 7 |
33 | fndm 5281 | . . . . . . 7 | |
34 | 32, 33 | syl 14 | . . . . . 6 |
35 | 25, 34 | eqtrd 2197 | . . . . 5 |
36 | 16, 35 | eleqtrrid 2254 | . . . 4 |
37 | fveq2 5480 | . . . . . . 7 | |
38 | 37 | dmeqd 4800 | . . . . . 6 |
39 | 38 | eleq2d 2234 | . . . . 5 |
40 | 39 | rspcev 2825 | . . . 4 |
41 | 14, 36, 40 | sylancr 411 | . . 3 |
42 | 17 | ad3antrrr 484 | . . . . . . . . 9 DECID |
43 | 18 | ad3antrrr 484 | . . . . . . . . 9 |
44 | 19 | ad3antrrr 484 | . . . . . . . . . 10 |
45 | fveq2 5480 | . . . . . . . . . . . . . 14 | |
46 | 45 | neeq1d 2352 | . . . . . . . . . . . . 13 |
47 | 46 | ralbidv 2464 | . . . . . . . . . . . 12 |
48 | 47 | cbvrexv 2690 | . . . . . . . . . . 11 |
49 | 48 | ralbii 2470 | . . . . . . . . . 10 |
50 | 44, 49 | sylib 121 | . . . . . . . . 9 |
51 | simplr 520 | . . . . . . . . 9 | |
52 | 42, 43, 50, 20, 21, 22, 23, 51 | ennnfonelemex 12290 | . . . . . . . 8 |
53 | 42 | ad2antrr 480 | . . . . . . . . . . . . . 14 DECID |
54 | 43 | ad2antrr 480 | . . . . . . . . . . . . . 14 |
55 | 44 | ad2antrr 480 | . . . . . . . . . . . . . 14 |
56 | simplr 520 | . . . . . . . . . . . . . 14 | |
57 | 53, 54, 55, 20, 21, 22, 23, 56 | ennnfonelemom 12284 | . . . . . . . . . . . . 13 |
58 | nnord 4583 | . . . . . . . . . . . . 13 | |
59 | 57, 58 | syl 14 | . . . . . . . . . . . 12 |
60 | simpr 109 | . . . . . . . . . . . 12 | |
61 | ordsucss 4475 | . . . . . . . . . . . 12 | |
62 | 59, 60, 61 | sylc 62 | . . . . . . . . . . 11 |
63 | simpr 109 | . . . . . . . . . . . . 13 | |
64 | 42, 43, 44, 20, 21, 22, 23, 51 | ennnfonelemom 12284 | . . . . . . . . . . . . . 14 |
65 | nnsucelsuc 6450 | . . . . . . . . . . . . . 14 | |
66 | 64, 65 | syl 14 | . . . . . . . . . . . . 13 |
67 | 63, 66 | mpbid 146 | . . . . . . . . . . . 12 |
68 | 67 | ad2antrr 480 | . . . . . . . . . . 11 |
69 | 62, 68 | sseldd 3138 | . . . . . . . . . 10 |
70 | 69 | ex 114 | . . . . . . . . 9 |
71 | 70 | reximdva 2566 | . . . . . . . 8 |
72 | 52, 71 | mpd 13 | . . . . . . 7 |
73 | 72 | rexlimdva2 2584 | . . . . . 6 |
74 | fveq2 5480 | . . . . . . . . 9 | |
75 | 74 | dmeqd 4800 | . . . . . . . 8 |
76 | 75 | eleq2d 2234 | . . . . . . 7 |
77 | 76 | cbvrexv 2690 | . . . . . 6 |
78 | 73, 77 | syl6ibr 161 | . . . . 5 |
79 | 78 | expcom 115 | . . . 4 |
80 | 79 | a2d 26 | . . 3 |
81 | 4, 7, 10, 13, 41, 80 | finds 4571 | . 2 |
82 | 1, 81 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 wral 2442 wrex 2443 cun 3109 wss 3111 c0 3404 cif 3515 csn 3570 cop 3573 cmpt 4037 word 4334 csuc 4337 com 4561 ccnv 4597 cdm 4598 cima 4601 wfn 5177 wf 5178 wfo 5180 cfv 5182 (class class class)co 5836 cmpo 5838 freccfrec 6349 cpm 6606 cc0 7744 c1 7745 caddc 7747 cmin 8060 cn0 9105 cz 9182 cseq 10370 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pm 6608 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-seqfrec 10371 |
This theorem is referenced by: ennnfonelemdm 12296 |
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