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Mirrors > Home > ILE Home > Th. List > ennnfonelemnn0 | Unicode version |
Description: Lemma for ennnfone 12301. A version of ennnfonelemen 12297 expressed in terms of instead of . (Contributed by Jim Kingdon, 27-Oct-2022.) |
Ref | Expression |
---|---|
ennnfonelemr.dceq | DECID |
ennnfonelemr.f | |
ennnfonelemr.n | |
ennnfonelemnn0.n | frec |
Ref | Expression |
---|---|
ennnfonelemnn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemr.dceq | . 2 DECID | |
2 | ennnfonelemr.f | . . 3 | |
3 | ennnfonelemnn0.n | . . . . . 6 frec | |
4 | 3 | frechashgf1o 10353 | . . . . 5 |
5 | f1ofo 5433 | . . . . 5 | |
6 | 4, 5 | ax-mp 5 | . . . 4 |
7 | 6 | a1i 9 | . . 3 |
8 | foco 5414 | . . 3 | |
9 | 2, 7, 8 | syl2anc 409 | . 2 |
10 | oveq2 5844 | . . . . . . 7 | |
11 | 10 | raleqdv 2665 | . . . . . 6 |
12 | 11 | rexbidv 2465 | . . . . 5 |
13 | ennnfonelemr.n | . . . . . 6 | |
14 | 13 | adantr 274 | . . . . 5 |
15 | f1of 5426 | . . . . . . . 8 | |
16 | 4, 15 | ax-mp 5 | . . . . . . 7 |
17 | 16 | a1i 9 | . . . . . 6 |
18 | simpr 109 | . . . . . 6 | |
19 | 17, 18 | ffvelrnd 5615 | . . . . 5 |
20 | 12, 14, 19 | rspcdva 2830 | . . . 4 |
21 | f1ocnv 5439 | . . . . . . . 8 | |
22 | f1of 5426 | . . . . . . . 8 | |
23 | 4, 21, 22 | mp2b 8 | . . . . . . 7 |
24 | 23 | a1i 9 | . . . . . 6 |
25 | simprl 521 | . . . . . 6 | |
26 | 24, 25 | ffvelrnd 5615 | . . . . 5 |
27 | fveq2 5480 | . . . . . . . . 9 | |
28 | 27 | neeq2d 2353 | . . . . . . . 8 |
29 | simplrr 526 | . . . . . . . 8 | |
30 | simpr 109 | . . . . . . . . . . 11 | |
31 | 18 | ad2antrr 480 | . . . . . . . . . . . 12 |
32 | peano2 4566 | . . . . . . . . . . . 12 | |
33 | 31, 32 | syl 14 | . . . . . . . . . . 11 |
34 | elnn 4577 | . . . . . . . . . . 11 | |
35 | 30, 33, 34 | syl2anc 409 | . . . . . . . . . 10 |
36 | 16 | ffvelrni 5613 | . . . . . . . . . 10 |
37 | 35, 36 | syl 14 | . . . . . . . . 9 |
38 | 0zd 9194 | . . . . . . . . . . . . 13 | |
39 | 38, 3, 35, 33 | frec2uzltd 10328 | . . . . . . . . . . . 12 |
40 | 30, 39 | mpd 13 | . . . . . . . . . . 11 |
41 | 38, 3, 31 | frec2uzsucd 10326 | . . . . . . . . . . 11 |
42 | 40, 41 | breqtrd 4002 | . . . . . . . . . 10 |
43 | 19 | ad2antrr 480 | . . . . . . . . . . 11 |
44 | nn0leltp1 9245 | . . . . . . . . . . 11 | |
45 | 37, 43, 44 | syl2anc 409 | . . . . . . . . . 10 |
46 | 42, 45 | mpbird 166 | . . . . . . . . 9 |
47 | fznn0 10038 | . . . . . . . . . 10 | |
48 | 43, 47 | syl 14 | . . . . . . . . 9 |
49 | 37, 46, 48 | mpbir2and 933 | . . . . . . . 8 |
50 | 28, 29, 49 | rspcdva 2830 | . . . . . . 7 |
51 | 26 | adantr 274 | . . . . . . . . 9 |
52 | fvco3 5551 | . . . . . . . . 9 | |
53 | 16, 51, 52 | sylancr 411 | . . . . . . . 8 |
54 | 25 | adantr 274 | . . . . . . . . . 10 |
55 | f1ocnvfv2 5740 | . . . . . . . . . 10 | |
56 | 4, 54, 55 | sylancr 411 | . . . . . . . . 9 |
57 | 56 | fveq2d 5484 | . . . . . . . 8 |
58 | 53, 57 | eqtrd 2197 | . . . . . . 7 |
59 | fvco3 5551 | . . . . . . . 8 | |
60 | 16, 35, 59 | sylancr 411 | . . . . . . 7 |
61 | 50, 58, 60 | 3netr4d 2367 | . . . . . 6 |
62 | 61 | ralrimiva 2537 | . . . . 5 |
63 | fveq2 5480 | . . . . . . . 8 | |
64 | 63 | neeq1d 2352 | . . . . . . 7 |
65 | 64 | ralbidv 2464 | . . . . . 6 |
66 | 65 | rspcev 2825 | . . . . 5 |
67 | 26, 62, 66 | syl2anc 409 | . . . 4 |
68 | 20, 67 | rexlimddv 2586 | . . 3 |
69 | 68 | ralrimiva 2537 | . 2 |
70 | id 19 | . . . 4 | |
71 | dmeq 4798 | . . . . . . 7 | |
72 | 71 | opeq1d 3758 | . . . . . 6 |
73 | 72 | sneqd 3583 | . . . . 5 |
74 | 70, 73 | uneq12d 3272 | . . . 4 |
75 | 70, 74 | ifeq12d 3534 | . . 3 |
76 | fveq2 5480 | . . . . 5 | |
77 | imaeq2 4936 | . . . . 5 | |
78 | 76, 77 | eleq12d 2235 | . . . 4 |
79 | 76 | opeq2d 3759 | . . . . . 6 |
80 | 79 | sneqd 3583 | . . . . 5 |
81 | 80 | uneq2d 3271 | . . . 4 |
82 | 78, 81 | ifbieq2d 3539 | . . 3 |
83 | 75, 82 | cbvmpov 5913 | . 2 |
84 | eqeq1 2171 | . . . 4 | |
85 | fvoveq1 5859 | . . . 4 | |
86 | 84, 85 | ifbieq2d 3539 | . . 3 |
87 | 86 | cbvmptv 4072 | . 2 |
88 | eqid 2164 | . 2 | |
89 | fveq2 5480 | . . 3 | |
90 | 89 | cbviunv 3899 | . 2 |
91 | 1, 9, 69, 83, 3, 87, 88, 90 | ennnfonelemen 12297 | 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 c0 3404 cif 3515 csn 3570 cop 3573 ciun 3860 class class class wbr 3976 cmpt 4037 csuc 4337 com 4561 ccnv 4597 cdm 4598 cima 4601 ccom 4602 wf 5178 wfo 5180 wf1o 5181 cfv 5182 (class class class)co 5836 cmpo 5838 freccfrec 6349 cpm 6606 cen 6695 cc0 7744 c1 7745 caddc 7747 clt 7924 cle 7925 cmin 8060 cn 8848 cn0 9105 cz 9182 cfz 9935 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-er 6492 df-pm 6608 df-en 6698 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-fz 9936 df-seqfrec 10371 |
This theorem is referenced by: ennnfonelemr 12299 |
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