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Mirrors > Home > ILE Home > Th. List > iseqf1olemnab | Unicode version |
Description: Lemma for seq3f1o 10232. (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 10215 | . . . . . 6 |
8 | 7 | adantr 274 | . . . . 5 |
9 | simprl 505 | . . . . . 6 | |
10 | 9 | iftrued 3451 | . . . . 5 |
11 | 8, 10 | eqtrd 2150 | . . . 4 |
12 | f1ocnvfv2 5647 | . . . . . . . 8 | |
13 | 4, 3, 12 | syl2anc 408 | . . . . . . 7 |
14 | 13 | ad2antrr 479 | . . . . . 6 |
15 | f1ofn 5336 | . . . . . . . . 9 | |
16 | 4, 15 | syl 14 | . . . . . . . 8 |
17 | 16 | ad2antrr 479 | . . . . . . 7 |
18 | elfzuz 9757 | . . . . . . . . . 10 | |
19 | fzss1 9798 | . . . . . . . . . 10 | |
20 | 3, 18, 19 | 3syl 17 | . . . . . . . . 9 |
21 | f1ocnv 5348 | . . . . . . . . . . . 12 | |
22 | f1of 5335 | . . . . . . . . . . . 12 | |
23 | 4, 21, 22 | 3syl 17 | . . . . . . . . . . 11 |
24 | 23, 3 | ffvelrnd 5524 | . . . . . . . . . 10 |
25 | elfzuz3 9758 | . . . . . . . . . 10 | |
26 | fzss2 9799 | . . . . . . . . . 10 | |
27 | 24, 25, 26 | 3syl 17 | . . . . . . . . 9 |
28 | 20, 27 | sstrd 3077 | . . . . . . . 8 |
29 | 28 | ad2antrr 479 | . . . . . . 7 |
30 | elfzubelfz 9771 | . . . . . . . . 9 | |
31 | 30 | adantr 274 | . . . . . . . 8 |
32 | 31 | ad2antlr 480 | . . . . . . 7 |
33 | fnfvima 5620 | . . . . . . 7 | |
34 | 17, 29, 32, 33 | syl3anc 1201 | . . . . . 6 |
35 | 14, 34 | eqeltrrd 2195 | . . . . 5 |
36 | 16 | ad2antrr 479 | . . . . . 6 |
37 | 28 | ad2antrr 479 | . . . . . 6 |
38 | 3 | adantr 274 | . . . . . . . . . 10 |
39 | elfzelz 9761 | . . . . . . . . . 10 | |
40 | 38, 39 | syl 14 | . . . . . . . . 9 |
41 | 40 | adantr 274 | . . . . . . . 8 |
42 | 24 | ad2antrr 479 | . . . . . . . . 9 |
43 | elfzelz 9761 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 5 | adantr 274 | . . . . . . . . . . 11 |
46 | elfzelz 9761 | . . . . . . . . . . 11 | |
47 | 45, 46 | syl 14 | . . . . . . . . . 10 |
48 | 47 | adantr 274 | . . . . . . . . 9 |
49 | peano2zm 9050 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 41, 44, 50 | 3jca 1146 | . . . . . . 7 |
52 | simpr 109 | . . . . . . . . . . 11 | |
53 | eqcom 2119 | . . . . . . . . . . 11 | |
54 | 52, 53 | sylnib 650 | . . . . . . . . . 10 |
55 | 9 | adantr 274 | . . . . . . . . . . . 12 |
56 | elfzle1 9762 | . . . . . . . . . . . 12 | |
57 | 55, 56 | syl 14 | . . . . . . . . . . 11 |
58 | zleloe 9059 | . . . . . . . . . . . 12 | |
59 | 41, 48, 58 | syl2anc 408 | . . . . . . . . . . 11 |
60 | 57, 59 | mpbid 146 | . . . . . . . . . 10 |
61 | 54, 60 | ecased 1312 | . . . . . . . . 9 |
62 | zltlem1 9069 | . . . . . . . . . 10 | |
63 | 41, 48, 62 | syl2anc 408 | . . . . . . . . 9 |
64 | 61, 63 | mpbid 146 | . . . . . . . 8 |
65 | 50 | zred 9131 | . . . . . . . . 9 |
66 | 48 | zred 9131 | . . . . . . . . 9 |
67 | 44 | zred 9131 | . . . . . . . . 9 |
68 | 66 | lem1d 8655 | . . . . . . . . 9 |
69 | elfzle2 9763 | . . . . . . . . . 10 | |
70 | 55, 69 | syl 14 | . . . . . . . . 9 |
71 | 65, 66, 67, 68, 70 | letrd 7854 | . . . . . . . 8 |
72 | 64, 71 | jca 304 | . . . . . . 7 |
73 | elfz2 9752 | . . . . . . 7 | |
74 | 51, 72, 73 | sylanbrc 413 | . . . . . 6 |
75 | fnfvima 5620 | . . . . . 6 | |
76 | 36, 37, 74, 75 | syl3anc 1201 | . . . . 5 |
77 | zdceq 9084 | . . . . . 6 DECID | |
78 | 47, 40, 77 | syl2anc 408 | . . . . 5 DECID |
79 | 35, 76, 78 | ifcldadc 3471 | . . . 4 |
80 | 11, 79 | eqeltrd 2194 | . . 3 |
81 | 2, 80 | eqeltrrd 2195 | . 2 |
82 | iseqf1olemnab.b | . . . . . 6 | |
83 | 3, 4, 82, 6 | iseqf1olemqval 10215 | . . . . 5 |
84 | 83 | adantr 274 | . . . 4 |
85 | simprr 506 | . . . . 5 | |
86 | 85 | iffalsed 3454 | . . . 4 |
87 | 84, 86 | eqtrd 2150 | . . 3 |
88 | f1of1 5334 | . . . . . . 7 | |
89 | 4, 88 | syl 14 | . . . . . 6 |
90 | f1elima 5642 | . . . . . 6 | |
91 | 89, 82, 28, 90 | syl3anc 1201 | . . . . 5 |
92 | 91 | adantr 274 | . . . 4 |
93 | 85, 92 | mtbird 647 | . . 3 |
94 | 87, 93 | eqneltrd 2213 | . 2 |
95 | 81, 94 | pm2.65da 635 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 DECID wdc 804 w3a 947 wceq 1316 wcel 1465 wss 3041 cif 3444 class class class wbr 3899 cmpt 3959 ccnv 4508 cima 4512 wfn 5088 wf 5089 wf1 5090 wf1o 5092 cfv 5093 (class class class)co 5742 c1 7589 clt 7768 cle 7769 cmin 7901 cz 9012 cuz 9282 cfz 9745 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-fz 9746 |
This theorem is referenced by: iseqf1olemmo 10220 |
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