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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 7285 | . . . . . . 7 | |
2 | 1 | brel 4638 | . . . . . 6 |
3 | 2 | simpld 111 | . . . . 5 |
4 | genpelvl.1 | . . . . . . . . 9 | |
5 | genpelvl.2 | . . . . . . . . 9 | |
6 | 4, 5 | genpelvl 7432 | . . . . . . . 8 |
7 | 6 | adantr 274 | . . . . . . 7 |
8 | breq2 3969 | . . . . . . . . . . . . 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 2581 | . . . . . . 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 1335 wcel 2128 wrex 2436 crab 2439 cop 3563 class class class wbr 3965 cfv 5170 (class class class)co 5824 cmpo 5826 c1st 6086 c2nd 6087 cnq 7200 cltq 7205 cnp 7211 |
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-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 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-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-id 4253 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-qs 6486 df-ni 7224 df-nqqs 7268 df-ltnqqs 7273 df-inp 7386 |
This theorem is referenced by: genprndl 7441 |
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