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Mirrors > Home > ILE Home > Th. List > prcunqu | Unicode version |
Description: An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.) |
Ref | Expression |
---|---|
prcunqu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7298 | . . . . . 6 | |
2 | 1 | brel 4651 | . . . . 5 |
3 | 2 | simprd 113 | . . . 4 |
4 | 3 | adantl 275 | . . 3 |
5 | breq2 3981 | . . . . . . 7 | |
6 | eleq1 2227 | . . . . . . 7 | |
7 | 5, 6 | imbi12d 233 | . . . . . 6 |
8 | 7 | imbi2d 229 | . . . . 5 |
9 | 1 | brel 4651 | . . . . . . . 8 |
10 | an42 577 | . . . . . . . . 9 | |
11 | breq1 3980 | . . . . . . . . . . . . . . . 16 | |
12 | eleq1 2227 | . . . . . . . . . . . . . . . 16 | |
13 | 11, 12 | anbi12d 465 | . . . . . . . . . . . . . . 15 |
14 | 13 | rspcev 2826 | . . . . . . . . . . . . . 14 |
15 | elinp 7407 | . . . . . . . . . . . . . . . 16 | |
16 | simpr1r 1044 | . . . . . . . . . . . . . . . 16 | |
17 | 15, 16 | sylbi 120 | . . . . . . . . . . . . . . 15 |
18 | 17 | r19.21bi 2552 | . . . . . . . . . . . . . 14 |
19 | 14, 18 | syl5ibrcom 156 | . . . . . . . . . . . . 13 |
20 | 19 | 3impb 1188 | . . . . . . . . . . . 12 |
21 | 20 | 3com12 1196 | . . . . . . . . . . 11 |
22 | 21 | 3expib 1195 | . . . . . . . . . 10 |
23 | 22 | impd 252 | . . . . . . . . 9 |
24 | 10, 23 | syl5bi 151 | . . . . . . . 8 |
25 | 9, 24 | mpand 426 | . . . . . . 7 |
26 | 25 | com12 30 | . . . . . 6 |
27 | 26 | ancoms 266 | . . . . 5 |
28 | 8, 27 | vtoclg 2782 | . . . 4 |
29 | 28 | impd 252 | . . 3 |
30 | 4, 29 | mpcom 36 | . 2 |
31 | 30 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wcel 2135 wral 2442 wrex 2443 wss 3112 cop 3574 class class class wbr 3977 cnq 7213 cltq 7218 cnp 7224 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-id 4266 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-qs 6499 df-ni 7237 df-nqqs 7281 df-ltnqqs 7286 df-inp 7399 |
This theorem is referenced by: prarloc 7436 prarloc2 7437 addnqprulem 7461 nqpru 7485 prmuloc2 7500 mulnqpru 7502 distrlem4pru 7518 1idpru 7524 ltexprlemm 7533 ltexprlemupu 7537 ltexprlemrl 7543 ltexprlemfu 7544 ltexprlemru 7545 aptiprlemu 7573 suplocexprlemdisj 7653 suplocexprlemub 7656 |
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