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Mirrors > Home > ILE Home > Th. List > ennnfonelemp1 | Unicode version |
Description: Lemma for ennnfone 12380. 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 9521 | . . . . 5 | |
3 | 1, 2 | eleqtrdi 2263 | . . . 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 12356 | . . . 4 |
12 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemg 12358 | . . . 4 |
13 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemjn 12357 | . . . 4 |
14 | 3, 11, 12, 13 | seqp1cd 10422 | . . 3 |
15 | 10 | fveq1i 5497 | . . . 4 |
16 | 15 | a1i 9 | . . 3 |
17 | 10 | fveq1i 5497 | . . . . 5 |
18 | 17 | a1i 9 | . . . 4 |
19 | eqeq1 2177 | . . . . . . 7 | |
20 | fvoveq1 5876 | . . . . . . 7 | |
21 | 19, 20 | ifbieq2d 3550 | . . . . . 6 |
22 | peano2nn0 9175 | . . . . . . 7 | |
23 | 1, 22 | syl 14 | . . . . . 6 |
24 | nn0p1gt0 9164 | . . . . . . . . . . . 12 | |
25 | 24 | gt0ne0d 8431 | . . . . . . . . . . 11 |
26 | 25 | neneqd 2361 | . . . . . . . . . 10 |
27 | 26 | iffalsed 3536 | . . . . . . . . 9 |
28 | nn0cn 9145 | . . . . . . . . . . 11 | |
29 | 1cnd 7936 | . . . . . . . . . . 11 | |
30 | 28, 29 | pncand 8231 | . . . . . . . . . 10 |
31 | 30 | fveq2d 5500 | . . . . . . . . 9 |
32 | 27, 31 | eqtrd 2203 | . . . . . . . 8 |
33 | 8 | frechashgf1o 10384 | . . . . . . . . . . 11 |
34 | f1ocnv 5455 | . . . . . . . . . . 11 | |
35 | 33, 34 | ax-mp 5 | . . . . . . . . . 10 |
36 | f1of 5442 | . . . . . . . . . 10 | |
37 | 35, 36 | mp1i 10 | . . . . . . . . 9 |
38 | id 19 | . . . . . . . . 9 | |
39 | 37, 38 | ffvelrnd 5632 | . . . . . . . 8 |
40 | 32, 39 | eqeltrd 2247 | . . . . . . 7 |
41 | 1, 40 | syl 14 | . . . . . 6 |
42 | 9, 21, 23, 41 | fvmptd3 5589 | . . . . 5 |
43 | 1, 32 | syl 14 | . . . . 5 |
44 | 42, 43 | eqtr2d 2204 | . . . 4 |
45 | 18, 44 | oveq12d 5871 | . . 3 |
46 | 14, 16, 45 | 3eqtr4d 2213 | . 2 |
47 | 4, 5, 6, 7, 8, 9, 10 | ennnfonelemh 12359 | . . . 4 |
48 | 47, 1 | ffvelrnd 5632 | . . 3 |
49 | 1, 39 | syl 14 | . . 3 |
50 | 48 | elexd 2743 | . . . 4 |
51 | dmexg 4875 | . . . . . . . 8 | |
52 | 50, 51 | syl 14 | . . . . . . 7 |
53 | fof 5420 | . . . . . . . . 9 | |
54 | 5, 53 | syl 14 | . . . . . . . 8 |
55 | 54, 49 | ffvelrnd 5632 | . . . . . . 7 |
56 | opexg 4213 | . . . . . . 7 | |
57 | 52, 55, 56 | syl2anc 409 | . . . . . 6 |
58 | snexg 4170 | . . . . . 6 | |
59 | 57, 58 | syl 14 | . . . . 5 |
60 | unexg 4428 | . . . . 5 | |
61 | 50, 59, 60 | syl2anc 409 | . . . 4 |
62 | 4, 5, 49 | ennnfonelemdc 12354 | . . . 4 DECID |
63 | 50, 61, 62 | ifcldcd 3561 | . . 3 |
64 | id 19 | . . . . 5 | |
65 | dmeq 4811 | . . . . . . . 8 | |
66 | 65 | opeq1d 3771 | . . . . . . 7 |
67 | 66 | sneqd 3596 | . . . . . 6 |
68 | 64, 67 | uneq12d 3282 | . . . . 5 |
69 | 64, 68 | ifeq12d 3545 | . . . 4 |
70 | fveq2 5496 | . . . . . 6 | |
71 | imaeq2 4949 | . . . . . 6 | |
72 | 70, 71 | eleq12d 2241 | . . . . 5 |
73 | 70 | opeq2d 3772 | . . . . . . 7 |
74 | 73 | sneqd 3596 | . . . . . 6 |
75 | 74 | uneq2d 3281 | . . . . 5 |
76 | 72, 75 | ifbieq2d 3550 | . . . 4 |
77 | 69, 76, 7 | ovmpog 5987 | . . 3 |
78 | 48, 49, 63, 77 | syl3anc 1233 | . 2 |
79 | 46, 78 | eqtrd 2203 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 DECID wdc 829 wceq 1348 wcel 2141 wne 2340 wral 2448 wrex 2449 crab 2452 cvv 2730 cun 3119 c0 3414 cif 3526 csn 3583 cop 3586 cmpt 4050 csuc 4350 com 4574 ccnv 4610 cdm 4611 cima 4614 wf 5194 wfo 5196 wf1o 5197 cfv 5198 (class class class)co 5853 cmpo 5855 freccfrec 6369 cpm 6627 cc0 7774 c1 7775 caddc 7777 cmin 8090 cn0 9135 cz 9212 cuz 9487 cseq 10401 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pm 6629 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 |
This theorem is referenced by: ennnfonelem1 12362 ennnfonelemhdmp1 12364 ennnfonelemss 12365 ennnfonelemkh 12367 ennnfonelemhf1o 12368 |
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