Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > genpelvu | Unicode version |
Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Ref | Expression |
---|---|
genpelvl.1 | |
genpelvl.2 |
Ref | Expression |
---|---|
genpelvu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genpelvl.1 | . . . . . . 7 | |
2 | genpelvl.2 | . . . . . . 7 | |
3 | 1, 2 | genipv 6699 | . . . . . 6 |
4 | 3 | fveq2d 5202 | . . . . 5 |
5 | nqex 6553 | . . . . . . 7 | |
6 | 5 | rabex 3922 | . . . . . 6 |
7 | 5 | rabex 3922 | . . . . . 6 |
8 | 6, 7 | op2nd 5794 | . . . . 5 |
9 | 4, 8 | syl6eq 2129 | . . . 4 |
10 | 9 | eleq2d 2148 | . . 3 |
11 | elrabi 2746 | . . 3 | |
12 | 10, 11 | syl6bi 161 | . 2 |
13 | prop 6665 | . . . . . . 7 | |
14 | elprnqu 6672 | . . . . . . 7 | |
15 | 13, 14 | sylan 277 | . . . . . 6 |
16 | prop 6665 | . . . . . . 7 | |
17 | elprnqu 6672 | . . . . . . 7 | |
18 | 16, 17 | sylan 277 | . . . . . 6 |
19 | 2 | caovcl 5675 | . . . . . 6 |
20 | 15, 18, 19 | syl2an 283 | . . . . 5 |
21 | 20 | an4s 552 | . . . 4 |
22 | eleq1 2141 | . . . 4 | |
23 | 21, 22 | syl5ibrcom 155 | . . 3 |
24 | 23 | rexlimdvva 2484 | . 2 |
25 | eqeq1 2087 | . . . . . 6 | |
26 | 25 | 2rexbidv 2391 | . . . . 5 |
27 | 26 | elrab3 2750 | . . . 4 |
28 | 10, 27 | sylan9bb 449 | . . 3 |
29 | 28 | ex 113 | . 2 |
30 | 12, 24, 29 | pm5.21ndd 653 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wrex 2349 crab 2352 cop 3401 cfv 4922 (class class class)co 5532 cmpt2 5534 c1st 5785 c2nd 5786 cnq 6470 cnp 6481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-qs 6135 df-ni 6494 df-nqqs 6538 df-inp 6656 |
This theorem is referenced by: genppreclu 6705 genpcuu 6710 genprndu 6712 genpdisj 6713 genpassu 6715 addnqprlemru 6748 mulnqprlemru 6764 distrlem1pru 6773 distrlem5pru 6777 1idpru 6781 ltexprlemfu 6801 recexprlem1ssu 6824 recexprlemss1u 6826 cauappcvgprlemladdfu 6844 caucvgprlemladdfu 6867 |
Copyright terms: Public domain | W3C validator |