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Mirrors > Home > ILE Home > Th. List > iseqf1olemnab | Unicode version |
Description: Lemma for seq3f1o 10429. (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 10412 | . . . . . 6 |
8 | 7 | adantr 274 | . . . . 5 |
9 | simprl 521 | . . . . . 6 | |
10 | 9 | iftrued 3522 | . . . . 5 |
11 | 8, 10 | eqtrd 2197 | . . . 4 |
12 | f1ocnvfv2 5740 | . . . . . . . 8 | |
13 | 4, 3, 12 | syl2anc 409 | . . . . . . 7 |
14 | 13 | ad2antrr 480 | . . . . . 6 |
15 | f1ofn 5427 | . . . . . . . . 9 | |
16 | 4, 15 | syl 14 | . . . . . . . 8 |
17 | 16 | ad2antrr 480 | . . . . . . 7 |
18 | elfzuz 9947 | . . . . . . . . . 10 | |
19 | fzss1 9988 | . . . . . . . . . 10 | |
20 | 3, 18, 19 | 3syl 17 | . . . . . . . . 9 |
21 | f1ocnv 5439 | . . . . . . . . . . . 12 | |
22 | f1of 5426 | . . . . . . . . . . . 12 | |
23 | 4, 21, 22 | 3syl 17 | . . . . . . . . . . 11 |
24 | 23, 3 | ffvelrnd 5615 | . . . . . . . . . 10 |
25 | elfzuz3 9948 | . . . . . . . . . 10 | |
26 | fzss2 9989 | . . . . . . . . . 10 | |
27 | 24, 25, 26 | 3syl 17 | . . . . . . . . 9 |
28 | 20, 27 | sstrd 3147 | . . . . . . . 8 |
29 | 28 | ad2antrr 480 | . . . . . . 7 |
30 | elfzubelfz 9961 | . . . . . . . . 9 | |
31 | 30 | adantr 274 | . . . . . . . 8 |
32 | 31 | ad2antlr 481 | . . . . . . 7 |
33 | fnfvima 5713 | . . . . . . 7 | |
34 | 17, 29, 32, 33 | syl3anc 1227 | . . . . . 6 |
35 | 14, 34 | eqeltrrd 2242 | . . . . 5 |
36 | 16 | ad2antrr 480 | . . . . . 6 |
37 | 28 | ad2antrr 480 | . . . . . 6 |
38 | 3 | adantr 274 | . . . . . . . . . 10 |
39 | elfzelz 9951 | . . . . . . . . . 10 | |
40 | 38, 39 | syl 14 | . . . . . . . . 9 |
41 | 40 | adantr 274 | . . . . . . . 8 |
42 | 24 | ad2antrr 480 | . . . . . . . . 9 |
43 | elfzelz 9951 | . . . . . . . . 9 | |
44 | 42, 43 | syl 14 | . . . . . . . 8 |
45 | 5 | adantr 274 | . . . . . . . . . . 11 |
46 | elfzelz 9951 | . . . . . . . . . . 11 | |
47 | 45, 46 | syl 14 | . . . . . . . . . 10 |
48 | 47 | adantr 274 | . . . . . . . . 9 |
49 | peano2zm 9220 | . . . . . . . . 9 | |
50 | 48, 49 | syl 14 | . . . . . . . 8 |
51 | 41, 44, 50 | 3jca 1166 | . . . . . . 7 |
52 | simpr 109 | . . . . . . . . . . 11 | |
53 | eqcom 2166 | . . . . . . . . . . 11 | |
54 | 52, 53 | sylnib 666 | . . . . . . . . . 10 |
55 | 9 | adantr 274 | . . . . . . . . . . . 12 |
56 | elfzle1 9952 | . . . . . . . . . . . 12 | |
57 | 55, 56 | syl 14 | . . . . . . . . . . 11 |
58 | zleloe 9229 | . . . . . . . . . . . 12 | |
59 | 41, 48, 58 | syl2anc 409 | . . . . . . . . . . 11 |
60 | 57, 59 | mpbid 146 | . . . . . . . . . 10 |
61 | 54, 60 | ecased 1338 | . . . . . . . . 9 |
62 | zltlem1 9239 | . . . . . . . . . 10 | |
63 | 41, 48, 62 | syl2anc 409 | . . . . . . . . 9 |
64 | 61, 63 | mpbid 146 | . . . . . . . 8 |
65 | 50 | zred 9304 | . . . . . . . . 9 |
66 | 48 | zred 9304 | . . . . . . . . 9 |
67 | 44 | zred 9304 | . . . . . . . . 9 |
68 | 66 | lem1d 8819 | . . . . . . . . 9 |
69 | elfzle2 9953 | . . . . . . . . . 10 | |
70 | 55, 69 | syl 14 | . . . . . . . . 9 |
71 | 65, 66, 67, 68, 70 | letrd 8013 | . . . . . . . 8 |
72 | 64, 71 | jca 304 | . . . . . . 7 |
73 | elfz2 9942 | . . . . . . 7 | |
74 | 51, 72, 73 | sylanbrc 414 | . . . . . 6 |
75 | fnfvima 5713 | . . . . . 6 | |
76 | 36, 37, 74, 75 | syl3anc 1227 | . . . . 5 |
77 | zdceq 9257 | . . . . . 6 DECID | |
78 | 47, 40, 77 | syl2anc 409 | . . . . 5 DECID |
79 | 35, 76, 78 | ifcldadc 3544 | . . . 4 |
80 | 11, 79 | eqeltrd 2241 | . . 3 |
81 | 2, 80 | eqeltrrd 2242 | . 2 |
82 | iseqf1olemnab.b | . . . . . 6 | |
83 | 3, 4, 82, 6 | iseqf1olemqval 10412 | . . . . 5 |
84 | 83 | adantr 274 | . . . 4 |
85 | simprr 522 | . . . . 5 | |
86 | 85 | iffalsed 3525 | . . . 4 |
87 | 84, 86 | eqtrd 2197 | . . 3 |
88 | f1of1 5425 | . . . . . . 7 | |
89 | 4, 88 | syl 14 | . . . . . 6 |
90 | f1elima 5735 | . . . . . 6 | |
91 | 89, 82, 28, 90 | syl3anc 1227 | . . . . 5 |
92 | 91 | adantr 274 | . . . 4 |
93 | 85, 92 | mtbird 663 | . . 3 |
94 | 87, 93 | eqneltrd 2260 | . 2 |
95 | 81, 94 | pm2.65da 651 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 824 w3a 967 wceq 1342 wcel 2135 wss 3111 cif 3515 class class class wbr 3976 cmpt 4037 ccnv 4597 cima 4601 wfn 5177 wf 5178 wf1 5179 wf1o 5181 cfv 5182 (class class class)co 5836 c1 7745 clt 7924 cle 7925 cmin 8060 cz 9182 cuz 9457 cfz 9935 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-fz 9936 |
This theorem is referenced by: iseqf1olemmo 10417 |
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