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Mirrors > Home > ILE Home > Th. List > ennnfonelemp1 | Unicode version |
Description: Lemma for ennnfone 12250. Value of at a successor. (Contributed by Jim Kingdon, 23-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemh.dceq | DECID |
ennnfonelemh.f | |
ennnfonelemh.ne | |
ennnfonelemh.g | |
ennnfonelemh.n | frec |
ennnfonelemh.j | |
ennnfonelemh.h | |
ennnfonelemp1.p |
Ref | Expression |
---|---|
ennnfonelemp1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemp1.p | . . . . 5 | |
2 | nn0uz 9479 | . . . . 5 | |
3 | 1, 2 | eleqtrdi 2250 | . . . 4 |
4 | ennnfonelemh.dceq | . . . . 5 DECID | |
5 | ennnfonelemh.f | . . . . 5 | |
6 | ennnfonelemh.ne | . . . . 5 | |
7 | ennnfonelemh.g | . . . . 5 | |
8 | ennnfonelemh.n | . . . . 5 frec | |
9 | ennnfonelemh.j | . . . . 5 | |
10 | ennnfonelemh.h | . . . . 5 | |
11 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemj0 12226 | . . . 4 |
12 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemg 12228 | . . . 4 |
13 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemjn 12227 | . . . 4 |
14 | 3, 11, 12, 13 | seqp1cd 10375 | . . 3 |
15 | 10 | fveq1i 5472 | . . . 4 |
16 | 15 | a1i 9 | . . 3 |
17 | 10 | fveq1i 5472 | . . . . 5 |
18 | 17 | a1i 9 | . . . 4 |
19 | eqeq1 2164 | . . . . . . 7 | |
20 | fvoveq1 5850 | . . . . . . 7 | |
21 | 19, 20 | ifbieq2d 3530 | . . . . . 6 |
22 | peano2nn0 9136 | . . . . . . 7 | |
23 | 1, 22 | syl 14 | . . . . . 6 |
24 | nn0p1gt0 9125 | . . . . . . . . . . . 12 | |
25 | 24 | gt0ne0d 8392 | . . . . . . . . . . 11 |
26 | 25 | neneqd 2348 | . . . . . . . . . 10 |
27 | 26 | iffalsed 3516 | . . . . . . . . 9 |
28 | nn0cn 9106 | . . . . . . . . . . 11 | |
29 | 1cnd 7897 | . . . . . . . . . . 11 | |
30 | 28, 29 | pncand 8192 | . . . . . . . . . 10 |
31 | 30 | fveq2d 5475 | . . . . . . . . 9 |
32 | 27, 31 | eqtrd 2190 | . . . . . . . 8 |
33 | 8 | frechashgf1o 10337 | . . . . . . . . . . 11 |
34 | f1ocnv 5430 | . . . . . . . . . . 11 | |
35 | 33, 34 | ax-mp 5 | . . . . . . . . . 10 |
36 | f1of 5417 | . . . . . . . . . 10 | |
37 | 35, 36 | mp1i 10 | . . . . . . . . 9 |
38 | id 19 | . . . . . . . . 9 | |
39 | 37, 38 | ffvelrnd 5606 | . . . . . . . 8 |
40 | 32, 39 | eqeltrd 2234 | . . . . . . 7 |
41 | 1, 40 | syl 14 | . . . . . 6 |
42 | 9, 21, 23, 41 | fvmptd3 5564 | . . . . 5 |
43 | 1, 32 | syl 14 | . . . . 5 |
44 | 42, 43 | eqtr2d 2191 | . . . 4 |
45 | 18, 44 | oveq12d 5845 | . . 3 |
46 | 14, 16, 45 | 3eqtr4d 2200 | . 2 |
47 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemh 12229 | . . . 4 |
48 | 47, 1 | ffvelrnd 5606 | . . 3 |
49 | 1, 39 | syl 14 | . . 3 |
50 | 48 | elexd 2725 | . . . 4 |
51 | dmexg 4853 | . . . . . . . 8 | |
52 | 50, 51 | syl 14 | . . . . . . 7 |
53 | fof 5395 | . . . . . . . . 9 | |
54 | 5, 53 | syl 14 | . . . . . . . 8 |
55 | 54, 49 | ffvelrnd 5606 | . . . . . . 7 |
56 | opexg 4191 | . . . . . . 7 | |
57 | 52, 55, 56 | syl2anc 409 | . . . . . 6 |
58 | snexg 4148 | . . . . . 6 | |
59 | 57, 58 | syl 14 | . . . . 5 |
60 | unexg 4406 | . . . . 5 | |
61 | 50, 59, 60 | syl2anc 409 | . . . 4 |
62 | 4, 5, 49 | ennnfonelemdc 12224 | . . . 4 DECID |
63 | 50, 61, 62 | ifcldcd 3541 | . . 3 |
64 | id 19 | . . . . 5 | |
65 | dmeq 4789 | . . . . . . . 8 | |
66 | 65 | opeq1d 3749 | . . . . . . 7 |
67 | 66 | sneqd 3574 | . . . . . 6 |
68 | 64, 67 | uneq12d 3263 | . . . . 5 |
69 | 64, 68 | ifeq12d 3525 | . . . 4 |
70 | fveq2 5471 | . . . . . 6 | |
71 | imaeq2 4927 | . . . . . 6 | |
72 | 70, 71 | eleq12d 2228 | . . . . 5 |
73 | 70 | opeq2d 3750 | . . . . . . 7 |
74 | 73 | sneqd 3574 | . . . . . 6 |
75 | 74 | uneq2d 3262 | . . . . 5 |
76 | 72, 75 | ifbieq2d 3530 | . . . 4 |
77 | 69, 76, 7 | ovmpog 5958 | . . 3 |
78 | 48, 49, 63, 77 | syl3anc 1220 | . 2 |
79 | 46, 78 | eqtrd 2190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 820 wceq 1335 wcel 2128 wne 2327 wral 2435 wrex 2436 crab 2439 cvv 2712 cun 3100 c0 3395 cif 3506 csn 3561 cop 3564 cmpt 4028 csuc 4328 com 4552 ccnv 4588 cdm 4589 cima 4592 wf 5169 wfo 5171 wf1o 5172 cfv 5173 (class class class)co 5827 cmpo 5829 freccfrec 6340 cpm 6597 cc0 7735 c1 7736 caddc 7738 cmin 8051 cn0 9096 cz 9173 cuz 9445 cseq 10354 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-ltadd 7851 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 df-recs 6255 df-frec 6341 df-pm 6599 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-uz 9446 df-seqfrec 10355 |
This theorem is referenced by: ennnfonelem1 12232 ennnfonelemhdmp1 12234 ennnfonelemss 12235 ennnfonelemkh 12237 ennnfonelemhf1o 12238 |
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