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Mirrors > Home > ILE Home > Th. List > prplnqu | Unicode version |
Description: Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.) |
Ref | Expression |
---|---|
prplnqu.x | |
prplnqu.q | |
prplnqu.sum |
Ref | Expression |
---|---|
prplnqu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prplnqu.q | . . . . . . . 8 | |
2 | nqprlu 6737 | . . . . . . . 8 | |
3 | 1, 2 | syl 14 | . . . . . . 7 |
4 | prplnqu.x | . . . . . . 7 | |
5 | ltaddpr 6787 | . . . . . . 7 | |
6 | 3, 4, 5 | syl2anc 403 | . . . . . 6 |
7 | addcomprg 6768 | . . . . . . 7 | |
8 | 3, 4, 7 | syl2anc 403 | . . . . . 6 |
9 | 6, 8 | breqtrd 3809 | . . . . 5 |
10 | prplnqu.sum | . . . . . 6 | |
11 | addclpr 6727 | . . . . . . . . 9 | |
12 | 4, 3, 11 | syl2anc 403 | . . . . . . . 8 |
13 | prop 6665 | . . . . . . . . 9 | |
14 | elprnqu 6672 | . . . . . . . . 9 | |
15 | 13, 14 | sylan 277 | . . . . . . . 8 |
16 | 12, 10, 15 | syl2anc 403 | . . . . . . 7 |
17 | nqpru 6742 | . . . . . . 7 | |
18 | 16, 12, 17 | syl2anc 403 | . . . . . 6 |
19 | 10, 18 | mpbid 145 | . . . . 5 |
20 | ltsopr 6786 | . . . . . 6 | |
21 | ltrelpr 6695 | . . . . . 6 | |
22 | 20, 21 | sotri 4740 | . . . . 5 |
23 | 9, 19, 22 | syl2anc 403 | . . . 4 |
24 | ltnqpr 6783 | . . . . 5 | |
25 | 1, 16, 24 | syl2anc 403 | . . . 4 |
26 | 23, 25 | mpbird 165 | . . 3 |
27 | ltexnqi 6599 | . . 3 | |
28 | 26, 27 | syl 14 | . 2 |
29 | 19 | adantr 270 | . . . . . 6 |
30 | 1 | adantr 270 | . . . . . . . . . 10 |
31 | simprl 497 | . . . . . . . . . 10 | |
32 | addcomnqg 6571 | . . . . . . . . . 10 | |
33 | 30, 31, 32 | syl2anc 403 | . . . . . . . . 9 |
34 | simprr 498 | . . . . . . . . 9 | |
35 | 33, 34 | eqtr3d 2115 | . . . . . . . 8 |
36 | breq2 3789 | . . . . . . . . . 10 | |
37 | 36 | abbidv 2196 | . . . . . . . . 9 |
38 | breq1 3788 | . . . . . . . . . 10 | |
39 | 38 | abbidv 2196 | . . . . . . . . 9 |
40 | 37, 39 | opeq12d 3578 | . . . . . . . 8 |
41 | 35, 40 | syl 14 | . . . . . . 7 |
42 | addnqpr 6751 | . . . . . . . 8 | |
43 | 31, 30, 42 | syl2anc 403 | . . . . . . 7 |
44 | 41, 43 | eqtr3d 2115 | . . . . . 6 |
45 | 29, 44 | breqtrd 3809 | . . . . 5 |
46 | ltaprg 6809 | . . . . . . 7 | |
47 | 46 | adantl 271 | . . . . . 6 |
48 | 4 | adantr 270 | . . . . . 6 |
49 | nqprlu 6737 | . . . . . . 7 | |
50 | 31, 49 | syl 14 | . . . . . 6 |
51 | 30, 2 | syl 14 | . . . . . 6 |
52 | addcomprg 6768 | . . . . . . 7 | |
53 | 52 | adantl 271 | . . . . . 6 |
54 | 47, 48, 50, 51, 53 | caovord2d 5690 | . . . . 5 |
55 | 45, 54 | mpbird 165 | . . . 4 |
56 | nqpru 6742 | . . . . 5 | |
57 | 31, 48, 56 | syl2anc 403 | . . . 4 |
58 | 55, 57 | mpbird 165 | . . 3 |
59 | oveq1 5539 | . . . . 5 | |
60 | 59 | eqeq1d 2089 | . . . 4 |
61 | 60 | rspcev 2701 | . . 3 |
62 | 58, 35, 61 | syl2anc 403 | . 2 |
63 | 28, 62 | 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 wrex 2349 cop 3401 class class class wbr 3785 cfv 4922 (class class class)co 5532 c1st 5785 c2nd 5786 cnq 6470 cplq 6472 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: caucvgprprlemexbt 6896 |
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