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Mirrors > Home > ILE Home > Th. List > caucvgprprlem2 | Unicode version |
Description: Lemma for caucvgprpr 7661. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.) |
Ref | Expression |
---|---|
caucvgprpr.f | |
caucvgprpr.cau | |
caucvgprpr.bnd | |
caucvgprpr.lim | |
caucvgprprlemlim.q | |
caucvgprprlemlim.jk | |
caucvgprprlemlim.jkq |
Ref | Expression |
---|---|
caucvgprprlem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprprlemlim.jk | . . . . 5 | |
2 | caucvgprprlemlim.jkq | . . . . 5 | |
3 | 1, 2 | caucvgprprlemk 7632 | . . . 4 |
4 | ltrelpi 7273 | . . . . . . . . . 10 | |
5 | 4 | brel 4661 | . . . . . . . . 9 |
6 | 1, 5 | syl 14 | . . . . . . . 8 |
7 | 6 | simprd 113 | . . . . . . 7 |
8 | nnnq 7371 | . . . . . . . 8 | |
9 | recclnq 7341 | . . . . . . . 8 | |
10 | 8, 9 | syl 14 | . . . . . . 7 |
11 | 7, 10 | syl 14 | . . . . . 6 |
12 | nqprlu 7496 | . . . . . 6 | |
13 | 11, 12 | syl 14 | . . . . 5 |
14 | caucvgprprlemlim.q | . . . . 5 | |
15 | caucvgprpr.f | . . . . . 6 | |
16 | 15, 7 | ffvelrnd 5629 | . . . . 5 |
17 | ltaprg 7568 | . . . . 5 | |
18 | 13, 14, 16, 17 | syl3anc 1233 | . . . 4 |
19 | 3, 18 | mpbid 146 | . . 3 |
20 | addclpr 7486 | . . . . 5 | |
21 | 16, 13, 20 | syl2anc 409 | . . . 4 |
22 | addclpr 7486 | . . . . 5 | |
23 | 16, 14, 22 | syl2anc 409 | . . . 4 |
24 | ltdfpr 7455 | . . . 4 | |
25 | 21, 23, 24 | syl2anc 409 | . . 3 |
26 | 19, 25 | mpbid 146 | . 2 |
27 | simprl 526 | . . . 4 | |
28 | 7 | adantr 274 | . . . . . 6 |
29 | simprrl 534 | . . . . . . . 8 | |
30 | breq1 3990 | . . . . . . . . . . . 12 | |
31 | 30 | cbvabv 2295 | . . . . . . . . . . 11 |
32 | breq2 3991 | . . . . . . . . . . . 12 | |
33 | 32 | cbvabv 2295 | . . . . . . . . . . 11 |
34 | 31, 33 | opeq12i 3768 | . . . . . . . . . 10 |
35 | 34 | oveq2i 5861 | . . . . . . . . 9 |
36 | 35 | fveq2i 5497 | . . . . . . . 8 |
37 | 29, 36 | eleqtrdi 2263 | . . . . . . 7 |
38 | nqprlu 7496 | . . . . . . . . . . 11 | |
39 | 11, 38 | syl 14 | . . . . . . . . . 10 |
40 | addclpr 7486 | . . . . . . . . . 10 | |
41 | 16, 39, 40 | syl2anc 409 | . . . . . . . . 9 |
42 | 41 | adantr 274 | . . . . . . . 8 |
43 | nqpru 7501 | . . . . . . . 8 | |
44 | 27, 42, 43 | syl2anc 409 | . . . . . . 7 |
45 | 37, 44 | mpbid 146 | . . . . . 6 |
46 | fveq2 5494 | . . . . . . . . 9 | |
47 | opeq1 3763 | . . . . . . . . . . . . . 14 | |
48 | 47 | eceq1d 6545 | . . . . . . . . . . . . 13 |
49 | 48 | fveq2d 5498 | . . . . . . . . . . . 12 |
50 | 49 | breq2d 3999 | . . . . . . . . . . 11 |
51 | 50 | abbidv 2288 | . . . . . . . . . 10 |
52 | 49 | breq1d 3997 | . . . . . . . . . . 11 |
53 | 52 | abbidv 2288 | . . . . . . . . . 10 |
54 | 51, 53 | opeq12d 3771 | . . . . . . . . 9 |
55 | 46, 54 | oveq12d 5868 | . . . . . . . 8 |
56 | 55 | breq1d 3997 | . . . . . . 7 |
57 | 56 | rspcev 2834 | . . . . . 6 |
58 | 28, 45, 57 | syl2anc 409 | . . . . 5 |
59 | breq2 3991 | . . . . . . . . . 10 | |
60 | 59 | abbidv 2288 | . . . . . . . . 9 |
61 | breq1 3990 | . . . . . . . . . 10 | |
62 | 61 | abbidv 2288 | . . . . . . . . 9 |
63 | 60, 62 | opeq12d 3771 | . . . . . . . 8 |
64 | 63 | breq2d 3999 | . . . . . . 7 |
65 | 64 | rexbidv 2471 | . . . . . 6 |
66 | caucvgprpr.lim | . . . . . . . 8 | |
67 | 66 | fveq2i 5497 | . . . . . . 7 |
68 | nqex 7312 | . . . . . . . . 9 | |
69 | 68 | rabex 4131 | . . . . . . . 8 |
70 | 68 | rabex 4131 | . . . . . . . 8 |
71 | 69, 70 | op2nd 6123 | . . . . . . 7 |
72 | 67, 71 | eqtri 2191 | . . . . . 6 |
73 | 65, 72 | elrab2 2889 | . . . . 5 |
74 | 27, 58, 73 | sylanbrc 415 | . . . 4 |
75 | simprrr 535 | . . . 4 | |
76 | rspe 2519 | . . . 4 | |
77 | 27, 74, 75, 76 | syl12anc 1231 | . . 3 |
78 | caucvgprpr.cau | . . . . . 6 | |
79 | caucvgprpr.bnd | . . . . . 6 | |
80 | 15, 78, 79, 66 | caucvgprprlemcl 7653 | . . . . 5 |
81 | 80 | adantr 274 | . . . 4 |
82 | 23 | adantr 274 | . . . 4 |
83 | ltdfpr 7455 | . . . 4 | |
84 | 81, 82, 83 | syl2anc 409 | . . 3 |
85 | 77, 84 | mpbird 166 | . 2 |
86 | 26, 85 | rexlimddv 2592 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 crab 2452 cop 3584 class class class wbr 3987 wf 5192 cfv 5196 (class class class)co 5850 c1st 6114 c2nd 6115 c1o 6385 cec 6507 cnpi 7221 clti 7224 ceq 7228 cnq 7229 cplq 7231 crq 7233 cltq 7234 cnp 7240 cpp 7242 cltp 7244 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-2o 6393 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-qs 6515 df-ni 7253 df-pli 7254 df-mi 7255 df-lti 7256 df-plpq 7293 df-mpq 7294 df-enq 7296 df-nqqs 7297 df-plqqs 7298 df-mqqs 7299 df-1nqqs 7300 df-rq 7301 df-ltnqqs 7302 df-enq0 7373 df-nq0 7374 df-0nq0 7375 df-plq0 7376 df-mq0 7377 df-inp 7415 df-iplp 7417 df-iltp 7419 |
This theorem is referenced by: caucvgprprlemlim 7660 |
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