Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > iseqf1olemnab | Unicode version |
Description: Lemma for seq3f1o 10284. (Contributed by Jim Kingdon, 27-Aug-2022.) |
Ref | Expression |
---|---|
iseqf1olemqcl.k | |
iseqf1olemqcl.j | |
iseqf1olemqcl.a | |
iseqf1olemnab.b | |
iseqf1olemnab.eq | |
iseqf1olemnab.q |
Ref | Expression |
---|---|
iseqf1olemnab |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqf1olemnab.eq | . . . 4 | |
2 | 1 | adantr 274 | . . 3 |
3 | iseqf1olemqcl.k | . . . . . . 7 | |
4 | iseqf1olemqcl.j | . . . . . . 7 | |
5 | iseqf1olemqcl.a | . . . . . . 7 | |
6 | iseqf1olemnab.q | . . . . . . 7 | |
7 | 3, 4, 5, 6 | iseqf1olemqval 10267 | . . . . . 6 |
8 | 7 | adantr 274 | . . . . 5 |
9 | simprl 520 | . . . . . 6 | |
10 | 9 | iftrued 3481 | . . . . 5 |
11 | 8, 10 | eqtrd 2172 | . . . 4 |
12 | f1ocnvfv2 5679 | . . . . . . . 8 | |
13 | 4, 3, 12 | syl2anc 408 | . . . . . . 7 |
14 | 13 | ad2antrr 479 | . . . . . 6 |
15 | f1ofn 5368 | . . . . . . . . 9 | |
16 | 4, 15 | syl 14 | . . . . . . . 8 |
17 | 16 | ad2antrr 479 | . . . . . . 7 |
18 | elfzuz 9809 | . . . . . . . . . 10 | |
19 | fzss1 9850 | . . . . . . . . . 10 | |
20 | 3, 18, 19 | 3syl 17 | . . . . . . . . 9 |
21 | f1ocnv 5380 | . . . . . . . . . . . 12 | |
22 | f1of 5367 | . . . . . . . . . . . 12 | |
23 | 4, 21, 22 | 3syl 17 | . . . . . . . . . . 11 |
24 | 23, 3 | ffvelrnd 5556 | . . . . . . . . . 10 |
25 | elfzuz3 9810 | . . . . . . . . . 10 | |
26 | fzss2 9851 | . . . . . . . . . 10 | |
27 | 24, 25, 26 | 3syl 17 | . . . . . . . . 9 |
28 | 20, 27 | sstrd 3107 | . . . . . . . 8 |
29 | 28 | ad2antrr 479 | . . . . . . 7 |
30 | elfzubelfz 9823 | . . . . . . . . 9 | |
31 | 30 | adantr 274 | . . . . . . . 8 |
32 | 31 | ad2antlr 480 | . . . . . . 7 |
33 | fnfvima 5652 | . . . . . . 7 | |
34 | 17, 29, 32, 33 | syl3anc 1216 | . . . . . 6 |
35 | 14, 34 | eqeltrrd 2217 | . . . . 5 |
36 | 16 | ad2antrr 479 | . . . . . 6 |
37 | 28 | ad2antrr 479 | . . . . . 6 |
38 | 3 | adantr 274 | . . . . . . . . . 10 |
39 | elfzelz 9813 | . . . . . . . . . 10 | |
40 | 38, 39 | syl 14 | . . . . . . . . 9 |
41 | 40 | adantr 274 | . . . . . . . 8 |
42 | 24 | ad2antrr 479 | . . . . . . . . 9 |
43 | elfzelz 9813 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 5 | adantr 274 | . . . . . . . . . . 11 |
46 | elfzelz 9813 | . . . . . . . . . . 11 | |
47 | 45, 46 | syl 14 | . . . . . . . . . 10 |
48 | 47 | adantr 274 | . . . . . . . . 9 |
49 | peano2zm 9099 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 41, 44, 50 | 3jca 1161 | . . . . . . 7 |
52 | simpr 109 | . . . . . . . . . . 11 | |
53 | eqcom 2141 | . . . . . . . . . . 11 | |
54 | 52, 53 | sylnib 665 | . . . . . . . . . 10 |
55 | 9 | adantr 274 | . . . . . . . . . . . 12 |
56 | elfzle1 9814 | . . . . . . . . . . . 12 | |
57 | 55, 56 | syl 14 | . . . . . . . . . . 11 |
58 | zleloe 9108 | . . . . . . . . . . . 12 | |
59 | 41, 48, 58 | syl2anc 408 | . . . . . . . . . . 11 |
60 | 57, 59 | mpbid 146 | . . . . . . . . . 10 |
61 | 54, 60 | ecased 1327 | . . . . . . . . 9 |
62 | zltlem1 9118 | . . . . . . . . . 10 | |
63 | 41, 48, 62 | syl2anc 408 | . . . . . . . . 9 |
64 | 61, 63 | mpbid 146 | . . . . . . . 8 |
65 | 50 | zred 9180 | . . . . . . . . 9 |
66 | 48 | zred 9180 | . . . . . . . . 9 |
67 | 44 | zred 9180 | . . . . . . . . 9 |
68 | 66 | lem1d 8698 | . . . . . . . . 9 |
69 | elfzle2 9815 | . . . . . . . . . 10 | |
70 | 55, 69 | syl 14 | . . . . . . . . 9 |
71 | 65, 66, 67, 68, 70 | letrd 7893 | . . . . . . . 8 |
72 | 64, 71 | jca 304 | . . . . . . 7 |
73 | elfz2 9804 | . . . . . . 7 | |
74 | 51, 72, 73 | sylanbrc 413 | . . . . . 6 |
75 | fnfvima 5652 | . . . . . 6 | |
76 | 36, 37, 74, 75 | syl3anc 1216 | . . . . 5 |
77 | zdceq 9133 | . . . . . 6 DECID | |
78 | 47, 40, 77 | syl2anc 408 | . . . . 5 DECID |
79 | 35, 76, 78 | ifcldadc 3501 | . . . 4 |
80 | 11, 79 | eqeltrd 2216 | . . 3 |
81 | 2, 80 | eqeltrrd 2217 | . 2 |
82 | iseqf1olemnab.b | . . . . . 6 | |
83 | 3, 4, 82, 6 | iseqf1olemqval 10267 | . . . . 5 |
84 | 83 | adantr 274 | . . . 4 |
85 | simprr 521 | . . . . 5 | |
86 | 85 | iffalsed 3484 | . . . 4 |
87 | 84, 86 | eqtrd 2172 | . . 3 |
88 | f1of1 5366 | . . . . . . 7 | |
89 | 4, 88 | syl 14 | . . . . . 6 |
90 | f1elima 5674 | . . . . . 6 | |
91 | 89, 82, 28, 90 | syl3anc 1216 | . . . . 5 |
92 | 91 | adantr 274 | . . . 4 |
93 | 85, 92 | mtbird 662 | . . 3 |
94 | 87, 93 | eqneltrd 2235 | . 2 |
95 | 81, 94 | pm2.65da 650 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 DECID wdc 819 w3a 962 wceq 1331 wcel 1480 wss 3071 cif 3474 class class class wbr 3929 cmpt 3989 ccnv 4538 cima 4542 wfn 5118 wf 5119 wf1 5120 wf1o 5122 cfv 5123 (class class class)co 5774 c1 7628 clt 7807 cle 7808 cmin 7940 cz 9061 cuz 9333 cfz 9797 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-addcom 7727 ax-addass 7729 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-0id 7735 ax-rnegex 7736 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-ltadd 7743 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-inn 8728 df-n0 8985 df-z 9062 df-uz 9334 df-fz 9798 |
This theorem is referenced by: iseqf1olemmo 10272 |
Copyright terms: Public domain | W3C validator |