Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > genpcdl | Unicode version |
Description: Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.) |
Ref | Expression |
---|---|
genpelvl.1 | |
genpelvl.2 | |
genpcdl.2 |
Ref | Expression |
---|---|
genpcdl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 7197 | . . . . . . 7 | |
2 | 1 | brel 4599 | . . . . . 6 |
3 | 2 | simpld 111 | . . . . 5 |
4 | genpelvl.1 | . . . . . . . . 9 | |
5 | genpelvl.2 | . . . . . . . . 9 | |
6 | 4, 5 | genpelvl 7344 | . . . . . . . 8 |
7 | 6 | adantr 274 | . . . . . . 7 |
8 | breq2 3941 | . . . . . . . . . . . . 13 | |
9 | 8 | biimpd 143 | . . . . . . . . . . . 12 |
10 | genpcdl.2 | . . . . . . . . . . . 12 | |
11 | 9, 10 | sylan9r 408 | . . . . . . . . . . 11 |
12 | 11 | exp31 362 | . . . . . . . . . 10 |
13 | 12 | an4s 578 | . . . . . . . . 9 |
14 | 13 | impancom 258 | . . . . . . . 8 |
15 | 14 | rexlimdvv 2559 | . . . . . . 7 |
16 | 7, 15 | sylbid 149 | . . . . . 6 |
17 | 16 | ex 114 | . . . . 5 |
18 | 3, 17 | syl5 32 | . . . 4 |
19 | 18 | com34 83 | . . 3 |
20 | 19 | pm2.43d 50 | . 2 |
21 | 20 | com23 78 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wceq 1332 wcel 1481 wrex 2418 crab 2421 cop 3535 class class class wbr 3937 cfv 5131 (class class class)co 5782 cmpo 5784 c1st 6044 c2nd 6045 cnq 7112 cltq 7117 cnp 7123 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-qs 6443 df-ni 7136 df-nqqs 7180 df-ltnqqs 7185 df-inp 7298 |
This theorem is referenced by: genprndl 7353 |
Copyright terms: Public domain | W3C validator |