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Mirrors > Home > ILE Home > Th. List > caucvgprprlemmu | Unicode version |
Description: Lemma for caucvgprpr 6902. The upper cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-Dec-2020.) |
Ref | Expression |
---|---|
caucvgprpr.f | |
caucvgprpr.cau | |
caucvgprpr.bnd | |
caucvgprpr.lim |
Ref | Expression |
---|---|
caucvgprprlemmu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprpr.f | . . . 4 | |
2 | 1pi 6505 | . . . . 5 | |
3 | 2 | a1i 9 | . . . 4 |
4 | 1, 3 | ffvelrnd 5324 | . . 3 |
5 | prop 6665 | . . 3 | |
6 | prmu 6668 | . . 3 | |
7 | 4, 5, 6 | 3syl 17 | . 2 |
8 | simprl 497 | . . . 4 | |
9 | 1nq 6556 | . . . 4 | |
10 | addclnq 6565 | . . . 4 | |
11 | 8, 9, 10 | sylancl 404 | . . 3 |
12 | 2 | a1i 9 | . . . . 5 |
13 | simprr 498 | . . . . . . . 8 | |
14 | 4 | adantr 270 | . . . . . . . . 9 |
15 | nqpru 6742 | . . . . . . . . 9 | |
16 | 8, 14, 15 | syl2anc 403 | . . . . . . . 8 |
17 | 13, 16 | mpbid 145 | . . . . . . 7 |
18 | ltaprg 6809 | . . . . . . . . 9 | |
19 | 18 | adantl 271 | . . . . . . . 8 |
20 | nqprlu 6737 | . . . . . . . . 9 | |
21 | 8, 20 | syl 14 | . . . . . . . 8 |
22 | nqprlu 6737 | . . . . . . . . 9 | |
23 | 9, 22 | mp1i 10 | . . . . . . . 8 |
24 | addcomprg 6768 | . . . . . . . . 9 | |
25 | 24 | adantl 271 | . . . . . . . 8 |
26 | 19, 14, 21, 23, 25 | caovord2d 5690 | . . . . . . 7 |
27 | 17, 26 | mpbid 145 | . . . . . 6 |
28 | df-1nqqs 6541 | . . . . . . . . . . . . 13 | |
29 | 28 | fveq2i 5201 | . . . . . . . . . . . 12 |
30 | rec1nq 6585 | . . . . . . . . . . . 12 | |
31 | 29, 30 | eqtr3i 2103 | . . . . . . . . . . 11 |
32 | 31 | breq2i 3793 | . . . . . . . . . 10 |
33 | 32 | abbii 2194 | . . . . . . . . 9 |
34 | 31 | breq1i 3792 | . . . . . . . . . 10 |
35 | 34 | abbii 2194 | . . . . . . . . 9 |
36 | 33, 35 | opeq12i 3575 | . . . . . . . 8 |
37 | 36 | oveq2i 5543 | . . . . . . 7 |
38 | 37 | a1i 9 | . . . . . 6 |
39 | addnqpr 6751 | . . . . . . 7 | |
40 | 8, 9, 39 | sylancl 404 | . . . . . 6 |
41 | 27, 38, 40 | 3brtr4d 3815 | . . . . 5 |
42 | fveq2 5198 | . . . . . . . 8 | |
43 | opeq1 3570 | . . . . . . . . . . . . 13 | |
44 | 43 | eceq1d 6165 | . . . . . . . . . . . 12 |
45 | 44 | fveq2d 5202 | . . . . . . . . . . 11 |
46 | 45 | breq2d 3797 | . . . . . . . . . 10 |
47 | 46 | abbidv 2196 | . . . . . . . . 9 |
48 | 45 | breq1d 3795 | . . . . . . . . . 10 |
49 | 48 | abbidv 2196 | . . . . . . . . 9 |
50 | 47, 49 | opeq12d 3578 | . . . . . . . 8 |
51 | 42, 50 | oveq12d 5550 | . . . . . . 7 |
52 | 51 | breq1d 3795 | . . . . . 6 |
53 | 52 | rspcev 2701 | . . . . 5 |
54 | 12, 41, 53 | syl2anc 403 | . . . 4 |
55 | breq2 3789 | . . . . . . . . 9 | |
56 | 55 | abbidv 2196 | . . . . . . . 8 |
57 | breq1 3788 | . . . . . . . . 9 | |
58 | 57 | abbidv 2196 | . . . . . . . 8 |
59 | 56, 58 | opeq12d 3578 | . . . . . . 7 |
60 | 59 | breq2d 3797 | . . . . . 6 |
61 | 60 | rexbidv 2369 | . . . . 5 |
62 | caucvgprpr.lim | . . . . . . 7 | |
63 | 62 | fveq2i 5201 | . . . . . 6 |
64 | nqex 6553 | . . . . . . . 8 | |
65 | 64 | rabex 3922 | . . . . . . 7 |
66 | 64 | rabex 3922 | . . . . . . 7 |
67 | 65, 66 | op2nd 5794 | . . . . . 6 |
68 | 63, 67 | eqtri 2101 | . . . . 5 |
69 | 61, 68 | elrab2 2751 | . . . 4 |
70 | 11, 54, 69 | sylanbrc 408 | . . 3 |
71 | eleq1 2141 | . . . 4 | |
72 | 71 | rspcev 2701 | . . 3 |
73 | 11, 70, 72 | syl2anc 403 | . 2 |
74 | 7, 73 | rexlimddv 2481 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 cab 2067 wral 2348 wrex 2349 crab 2352 cop 3401 class class class wbr 3785 wf 4918 cfv 4922 (class class class)co 5532 c1st 5785 c2nd 5786 c1o 6017 cec 6127 cnpi 6462 clti 6465 ceq 6469 cnq 6470 c1q 6471 cplq 6472 crq 6474 cltq 6475 cnp 6481 cpp 6483 cltp 6485 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-iltp 6660 |
This theorem is referenced by: caucvgprprlemm 6886 |
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