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Mirrors > Home > ILE Home > Th. List > caucvgprlemcl | Unicode version |
Description: Lemma for caucvgpr 6872. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.) |
Ref | Expression |
---|---|
caucvgpr.f | |
caucvgpr.cau | |
caucvgpr.bnd | |
caucvgpr.lim |
Ref | Expression |
---|---|
caucvgprlemcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgpr.f | . . . 4 | |
2 | caucvgpr.cau | . . . 4 | |
3 | caucvgpr.bnd | . . . . 5 | |
4 | fveq2 5198 | . . . . . . 7 | |
5 | 4 | breq2d 3797 | . . . . . 6 |
6 | 5 | cbvralv 2577 | . . . . 5 |
7 | 3, 6 | sylib 120 | . . . 4 |
8 | caucvgpr.lim | . . . . 5 | |
9 | opeq1 3570 | . . . . . . . . . . . . 13 | |
10 | 9 | eceq1d 6165 | . . . . . . . . . . . 12 |
11 | 10 | fveq2d 5202 | . . . . . . . . . . 11 |
12 | 11 | oveq2d 5548 | . . . . . . . . . 10 |
13 | 12, 4 | breq12d 3798 | . . . . . . . . 9 |
14 | 13 | cbvrexv 2578 | . . . . . . . 8 |
15 | 14 | a1i 9 | . . . . . . 7 |
16 | 15 | rabbiia 2591 | . . . . . 6 |
17 | 4, 11 | oveq12d 5550 | . . . . . . . . . 10 |
18 | 17 | breq1d 3795 | . . . . . . . . 9 |
19 | 18 | cbvrexv 2578 | . . . . . . . 8 |
20 | 19 | a1i 9 | . . . . . . 7 |
21 | 20 | rabbiia 2591 | . . . . . 6 |
22 | 16, 21 | opeq12i 3575 | . . . . 5 |
23 | 8, 22 | eqtri 2101 | . . . 4 |
24 | 1, 2, 7, 23 | caucvgprlemm 6858 | . . 3 |
25 | ssrab2 3079 | . . . . . 6 | |
26 | nqex 6553 | . . . . . . 7 | |
27 | 26 | elpw2 3932 | . . . . . 6 |
28 | 25, 27 | mpbir 144 | . . . . 5 |
29 | ssrab2 3079 | . . . . . 6 | |
30 | 26 | elpw2 3932 | . . . . . 6 |
31 | 29, 30 | mpbir 144 | . . . . 5 |
32 | opelxpi 4394 | . . . . 5 | |
33 | 28, 31, 32 | mp2an 416 | . . . 4 |
34 | 8, 33 | eqeltri 2151 | . . 3 |
35 | 24, 34 | jctil 305 | . 2 |
36 | 1, 2, 7, 23 | caucvgprlemrnd 6863 | . . 3 |
37 | breq1 3788 | . . . . . . 7 | |
38 | fveq2 5198 | . . . . . . . . 9 | |
39 | opeq1 3570 | . . . . . . . . . . . 12 | |
40 | 39 | eceq1d 6165 | . . . . . . . . . . 11 |
41 | 40 | fveq2d 5202 | . . . . . . . . . 10 |
42 | 41 | oveq2d 5548 | . . . . . . . . 9 |
43 | 38, 42 | breq12d 3798 | . . . . . . . 8 |
44 | 38, 41 | oveq12d 5550 | . . . . . . . . 9 |
45 | 44 | breq2d 3797 | . . . . . . . 8 |
46 | 43, 45 | anbi12d 456 | . . . . . . 7 |
47 | 37, 46 | imbi12d 232 | . . . . . 6 |
48 | breq2 3789 | . . . . . . 7 | |
49 | fveq2 5198 | . . . . . . . . . 10 | |
50 | 49 | oveq1d 5547 | . . . . . . . . 9 |
51 | 50 | breq2d 3797 | . . . . . . . 8 |
52 | 49 | breq1d 3795 | . . . . . . . 8 |
53 | 51, 52 | anbi12d 456 | . . . . . . 7 |
54 | 48, 53 | imbi12d 232 | . . . . . 6 |
55 | 47, 54 | cbvral2v 2585 | . . . . 5 |
56 | 2, 55 | sylib 120 | . . . 4 |
57 | 1, 56, 7, 23 | caucvgprlemdisj 6864 | . . 3 |
58 | 1, 2, 7, 23 | caucvgprlemloc 6865 | . . 3 |
59 | 36, 57, 58 | 3jca 1118 | . 2 |
60 | elnp1st2nd 6666 | . 2 | |
61 | 35, 59, 60 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 wral 2348 wrex 2349 crab 2352 wss 2973 cpw 3382 cop 3401 class class class wbr 3785 cxp 4361 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 cplq 6472 crq 6474 cltq 6475 cnp 6481 |
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-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-inp 6656 |
This theorem is referenced by: caucvgprlemladdfu 6867 caucvgprlemladdrl 6868 caucvgprlem1 6869 caucvgprlem2 6870 caucvgpr 6872 |
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