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Mirrors > Home > MPE Home > Th. List > genpcd | Structured version Visualization version Unicode version |
Description: Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
genp.1 | |
genp.2 | |
genpcd.2 |
Ref | Expression |
---|---|
genpcd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrelnq 9748 | . . . . . . 7 | |
2 | 1 | brel 5168 | . . . . . 6 |
3 | 2 | simpld 475 | . . . . 5 |
4 | genp.1 | . . . . . . . . 9 | |
5 | genp.2 | . . . . . . . . 9 | |
6 | 4, 5 | genpelv 9822 | . . . . . . . 8 |
7 | 6 | adantr 481 | . . . . . . 7 |
8 | breq2 4657 | . . . . . . . . . . . . 13 | |
9 | 8 | biimpd 219 | . . . . . . . . . . . 12 |
10 | genpcd.2 | . . . . . . . . . . . 12 | |
11 | 9, 10 | sylan9r 690 | . . . . . . . . . . 11 |
12 | 11 | exp31 630 | . . . . . . . . . 10 |
13 | 12 | an4s 869 | . . . . . . . . 9 |
14 | 13 | impancom 456 | . . . . . . . 8 |
15 | 14 | rexlimdvv 3037 | . . . . . . 7 |
16 | 7, 15 | sylbid 230 | . . . . . 6 |
17 | 16 | ex 450 | . . . . 5 |
18 | 3, 17 | syl5 34 | . . . 4 |
19 | 18 | com34 91 | . . 3 |
20 | 19 | pm2.43d 53 | . 2 |
21 | 20 | com23 86 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cab 2608 wrex 2913 class class class wbr 4653 (class class class)co 6650 cmpt2 6652 cnq 9674 cltq 9680 cnp 9681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-ni 9694 df-nq 9734 df-ltnq 9740 df-np 9803 |
This theorem is referenced by: genpcl 9830 |
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