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Mirrors > Home > ILE Home > Th. List > caucvgprprlemopu | Unicode version |
Description: Lemma for caucvgprpr 7632. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-2020.) |
Ref | Expression |
---|---|
caucvgprpr.f | |
caucvgprpr.cau | |
caucvgprpr.bnd | |
caucvgprpr.lim |
Ref | Expression |
---|---|
caucvgprprlemopu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caucvgprpr.lim | . . . . 5 | |
2 | 1 | caucvgprprlemelu 7606 | . . . 4 |
3 | 2 | simprbi 273 | . . 3 |
4 | 3 | adantl 275 | . 2 |
5 | simprr 522 | . . . . 5 | |
6 | caucvgprpr.f | . . . . . . . . 9 | |
7 | 6 | ffvelrnda 5602 | . . . . . . . 8 |
8 | recnnpr 7468 | . . . . . . . . 9 | |
9 | 8 | adantl 275 | . . . . . . . 8 |
10 | addclpr 7457 | . . . . . . . 8 | |
11 | 7, 9, 10 | syl2anc 409 | . . . . . . 7 |
12 | 11 | ad2ant2r 501 | . . . . . 6 |
13 | 2 | simplbi 272 | . . . . . . . 8 |
14 | 13 | ad2antlr 481 | . . . . . . 7 |
15 | nqprlu 7467 | . . . . . . 7 | |
16 | 14, 15 | syl 14 | . . . . . 6 |
17 | ltdfpr 7426 | . . . . . 6 | |
18 | 12, 16, 17 | syl2anc 409 | . . . . 5 |
19 | 5, 18 | mpbid 146 | . . . 4 |
20 | simpr 109 | . . . . . . . 8 | |
21 | 12 | adantr 274 | . . . . . . . 8 |
22 | nqpru 7472 | . . . . . . . 8 | |
23 | 20, 21, 22 | syl2anc 409 | . . . . . . 7 |
24 | vex 2715 | . . . . . . . . 9 | |
25 | breq1 3968 | . . . . . . . . 9 | |
26 | ltnqex 7469 | . . . . . . . . . 10 | |
27 | gtnqex 7470 | . . . . . . . . . 10 | |
28 | 26, 27 | op1st 6094 | . . . . . . . . 9 |
29 | 24, 25, 28 | elab2 2860 | . . . . . . . 8 |
30 | 29 | a1i 9 | . . . . . . 7 |
31 | 23, 30 | anbi12d 465 | . . . . . 6 |
32 | 31 | biimpd 143 | . . . . 5 |
33 | 32 | reximdva 2559 | . . . 4 |
34 | 19, 33 | mpd 13 | . . 3 |
35 | simprr 522 | . . . . . 6 | |
36 | simplr 520 | . . . . . . 7 | |
37 | simplrl 525 | . . . . . . . . 9 | |
38 | 37 | adantr 274 | . . . . . . . 8 |
39 | simprl 521 | . . . . . . . 8 | |
40 | fveq2 5468 | . . . . . . . . . . 11 | |
41 | opeq1 3741 | . . . . . . . . . . . . . . . 16 | |
42 | 41 | eceq1d 6516 | . . . . . . . . . . . . . . 15 |
43 | 42 | fveq2d 5472 | . . . . . . . . . . . . . 14 |
44 | 43 | breq2d 3977 | . . . . . . . . . . . . 13 |
45 | 44 | abbidv 2275 | . . . . . . . . . . . 12 |
46 | 43 | breq1d 3975 | . . . . . . . . . . . . 13 |
47 | 46 | abbidv 2275 | . . . . . . . . . . . 12 |
48 | 45, 47 | opeq12d 3749 | . . . . . . . . . . 11 |
49 | 40, 48 | oveq12d 5842 | . . . . . . . . . 10 |
50 | 49 | breq1d 3975 | . . . . . . . . 9 |
51 | 50 | rspcev 2816 | . . . . . . . 8 |
52 | 38, 39, 51 | syl2anc 409 | . . . . . . 7 |
53 | 1 | caucvgprprlemelu 7606 | . . . . . . 7 |
54 | 36, 52, 53 | sylanbrc 414 | . . . . . 6 |
55 | 35, 54 | jca 304 | . . . . 5 |
56 | 55 | ex 114 | . . . 4 |
57 | 56 | reximdva 2559 | . . 3 |
58 | 34, 57 | mpd 13 | . 2 |
59 | 4, 58 | rexlimddv 2579 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1335 wcel 2128 cab 2143 wral 2435 wrex 2436 crab 2439 cop 3563 class class class wbr 3965 wf 5166 cfv 5170 (class class class)co 5824 c1st 6086 c2nd 6087 c1o 6356 cec 6478 cnpi 7192 clti 7195 ceq 7199 cnq 7200 cplq 7202 crq 7204 cltq 7205 cnp 7211 cpp 7213 cltp 7215 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-iplp 7388 df-iltp 7390 |
This theorem is referenced by: caucvgprprlemrnd 7621 |
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