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Mirrors > Home > ILE Home > Th. List > iseqf1olemnab | Unicode version |
Description: Lemma for seq3f1o 10474. (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 276 | . . 3 |
3 | iseqf1olemqcl.k | . . . . . . 7 | |
4 | iseqf1olemqcl.j | . . . . . . 7 | |
5 | iseqf1olemqcl.a | . . . . . . 7 | |
6 | iseqf1olemnab.q | . . . . . . 7 | |
7 | 3, 4, 5, 6 | iseqf1olemqval 10457 | . . . . . 6 |
8 | 7 | adantr 276 | . . . . 5 |
9 | simprl 529 | . . . . . 6 | |
10 | 9 | iftrued 3539 | . . . . 5 |
11 | 8, 10 | eqtrd 2208 | . . . 4 |
12 | f1ocnvfv2 5769 | . . . . . . . 8 | |
13 | 4, 3, 12 | syl2anc 411 | . . . . . . 7 |
14 | 13 | ad2antrr 488 | . . . . . 6 |
15 | f1ofn 5454 | . . . . . . . . 9 | |
16 | 4, 15 | syl 14 | . . . . . . . 8 |
17 | 16 | ad2antrr 488 | . . . . . . 7 |
18 | elfzuz 9991 | . . . . . . . . . 10 | |
19 | fzss1 10033 | . . . . . . . . . 10 | |
20 | 3, 18, 19 | 3syl 17 | . . . . . . . . 9 |
21 | f1ocnv 5466 | . . . . . . . . . . . 12 | |
22 | f1of 5453 | . . . . . . . . . . . 12 | |
23 | 4, 21, 22 | 3syl 17 | . . . . . . . . . . 11 |
24 | 23, 3 | ffvelcdmd 5644 | . . . . . . . . . 10 |
25 | elfzuz3 9992 | . . . . . . . . . 10 | |
26 | fzss2 10034 | . . . . . . . . . 10 | |
27 | 24, 25, 26 | 3syl 17 | . . . . . . . . 9 |
28 | 20, 27 | sstrd 3163 | . . . . . . . 8 |
29 | 28 | ad2antrr 488 | . . . . . . 7 |
30 | elfzubelfz 10006 | . . . . . . . . 9 | |
31 | 30 | adantr 276 | . . . . . . . 8 |
32 | 31 | ad2antlr 489 | . . . . . . 7 |
33 | fnfvima 5742 | . . . . . . 7 | |
34 | 17, 29, 32, 33 | syl3anc 1238 | . . . . . 6 |
35 | 14, 34 | eqeltrrd 2253 | . . . . 5 |
36 | 16 | ad2antrr 488 | . . . . . 6 |
37 | 28 | ad2antrr 488 | . . . . . 6 |
38 | 3 | adantr 276 | . . . . . . . . . 10 |
39 | elfzelz 9995 | . . . . . . . . . 10 | |
40 | 38, 39 | syl 14 | . . . . . . . . 9 |
41 | 40 | adantr 276 | . . . . . . . 8 |
42 | 24 | ad2antrr 488 | . . . . . . . . 9 |
43 | elfzelz 9995 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 5 | adantr 276 | . . . . . . . . . . 11 |
46 | elfzelz 9995 | . . . . . . . . . . 11 | |
47 | 45, 46 | syl 14 | . . . . . . . . . 10 |
48 | 47 | adantr 276 | . . . . . . . . 9 |
49 | peano2zm 9264 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 41, 44, 50 | 3jca 1177 | . . . . . . 7 |
52 | simpr 110 | . . . . . . . . . . 11 | |
53 | eqcom 2177 | . . . . . . . . . . 11 | |
54 | 52, 53 | sylnib 676 | . . . . . . . . . 10 |
55 | 9 | adantr 276 | . . . . . . . . . . . 12 |
56 | elfzle1 9997 | . . . . . . . . . . . 12 | |
57 | 55, 56 | syl 14 | . . . . . . . . . . 11 |
58 | zleloe 9273 | . . . . . . . . . . . 12 | |
59 | 41, 48, 58 | syl2anc 411 | . . . . . . . . . . 11 |
60 | 57, 59 | mpbid 147 | . . . . . . . . . 10 |
61 | 54, 60 | ecased 1349 | . . . . . . . . 9 |
62 | zltlem1 9283 | . . . . . . . . . 10 | |
63 | 41, 48, 62 | syl2anc 411 | . . . . . . . . 9 |
64 | 61, 63 | mpbid 147 | . . . . . . . 8 |
65 | 50 | zred 9348 | . . . . . . . . 9 |
66 | 48 | zred 9348 | . . . . . . . . 9 |
67 | 44 | zred 9348 | . . . . . . . . 9 |
68 | 66 | lem1d 8863 | . . . . . . . . 9 |
69 | elfzle2 9998 | . . . . . . . . . 10 | |
70 | 55, 69 | syl 14 | . . . . . . . . 9 |
71 | 65, 66, 67, 68, 70 | letrd 8055 | . . . . . . . 8 |
72 | 64, 71 | jca 306 | . . . . . . 7 |
73 | elfz2 9986 | . . . . . . 7 | |
74 | 51, 72, 73 | sylanbrc 417 | . . . . . 6 |
75 | fnfvima 5742 | . . . . . 6 | |
76 | 36, 37, 74, 75 | syl3anc 1238 | . . . . 5 |
77 | zdceq 9301 | . . . . . 6 DECID | |
78 | 47, 40, 77 | syl2anc 411 | . . . . 5 DECID |
79 | 35, 76, 78 | ifcldadc 3561 | . . . 4 |
80 | 11, 79 | eqeltrd 2252 | . . 3 |
81 | 2, 80 | eqeltrrd 2253 | . 2 |
82 | iseqf1olemnab.b | . . . . . 6 | |
83 | 3, 4, 82, 6 | iseqf1olemqval 10457 | . . . . 5 |
84 | 83 | adantr 276 | . . . 4 |
85 | simprr 531 | . . . . 5 | |
86 | 85 | iffalsed 3542 | . . . 4 |
87 | 84, 86 | eqtrd 2208 | . . 3 |
88 | f1of1 5452 | . . . . . . 7 | |
89 | 4, 88 | syl 14 | . . . . . 6 |
90 | f1elima 5764 | . . . . . 6 | |
91 | 89, 82, 28, 90 | syl3anc 1238 | . . . . 5 |
92 | 91 | adantr 276 | . . . 4 |
93 | 85, 92 | mtbird 673 | . . 3 |
94 | 87, 93 | eqneltrd 2271 | . 2 |
95 | 81, 94 | pm2.65da 661 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 104 wb 105 wo 708 DECID wdc 834 w3a 978 wceq 1353 wcel 2146 wss 3127 cif 3532 class class class wbr 3998 cmpt 4059 ccnv 4619 cima 4623 wfn 5203 wf 5204 wf1 5205 wf1o 5207 cfv 5208 (class class class)co 5865 c1 7787 clt 7966 cle 7967 cmin 8102 cz 9226 cuz 9501 cfz 9979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-ltadd 7902 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8893 df-n0 9150 df-z 9227 df-uz 9502 df-fz 9980 |
This theorem is referenced by: iseqf1olemmo 10462 |
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