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Mirrors > Home > ILE Home > Th. List > prcdnql | Unicode version |
Description: A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.) |
Ref | Expression |
---|---|
prcdnql |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7173 | . . . . . 6 | |
2 | 1 | brel 4591 | . . . . 5 |
3 | 2 | simpld 111 | . . . 4 |
4 | 3 | adantl 275 | . . 3 |
5 | breq1 3932 | . . . . . . 7 | |
6 | eleq1 2202 | . . . . . . 7 | |
7 | 5, 6 | imbi12d 233 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | 1 | brel 4591 | . . . . . . . . 9 |
10 | 9 | ancomd 265 | . . . . . . . 8 |
11 | an42 576 | . . . . . . . . 9 | |
12 | breq2 3933 | . . . . . . . . . . . . . . . 16 | |
13 | eleq1 2202 | . . . . . . . . . . . . . . . 16 | |
14 | 12, 13 | anbi12d 464 | . . . . . . . . . . . . . . 15 |
15 | 14 | rspcev 2789 | . . . . . . . . . . . . . 14 |
16 | elinp 7282 | . . . . . . . . . . . . . . . 16 | |
17 | simpr1l 1038 | . . . . . . . . . . . . . . . 16 | |
18 | 16, 17 | sylbi 120 | . . . . . . . . . . . . . . 15 |
19 | 18 | r19.21bi 2520 | . . . . . . . . . . . . . 14 |
20 | 15, 19 | syl5ibrcom 156 | . . . . . . . . . . . . 13 |
21 | 20 | 3impb 1177 | . . . . . . . . . . . 12 |
22 | 21 | 3com12 1185 | . . . . . . . . . . 11 |
23 | 22 | 3expib 1184 | . . . . . . . . . 10 |
24 | 23 | impd 252 | . . . . . . . . 9 |
25 | 11, 24 | syl5bi 151 | . . . . . . . 8 |
26 | 10, 25 | mpand 425 | . . . . . . 7 |
27 | 26 | com12 30 | . . . . . 6 |
28 | 27 | ancoms 266 | . . . . 5 |
29 | 8, 28 | vtoclg 2746 | . . . 4 |
30 | 29 | impd 252 | . . 3 |
31 | 4, 30 | mpcom 36 | . 2 |
32 | 31 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wral 2416 wrex 2417 wss 3071 cop 3530 class class class wbr 3929 cnq 7088 cltq 7093 cnp 7099 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-qs 6435 df-ni 7112 df-nqqs 7156 df-ltnqqs 7161 df-inp 7274 |
This theorem is referenced by: prubl 7294 addnqprllem 7335 nqprl 7359 mulnqprl 7376 distrlem4prl 7392 ltprordil 7397 1idprl 7398 ltpopr 7403 ltaddpr 7405 ltexprlemlol 7410 ltexprlemfl 7417 ltexprlemrl 7418 aptiprleml 7447 aptiprlemu 7448 archrecpr 7472 caucvgprprlemml 7502 suplocexprlemrl 7525 suplocexprlemloc 7529 |
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