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Mirrors > Home > ILE Home > Th. List > genpmu | Unicode version |
Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Ref | Expression |
---|---|
genpelvl.1 | |
genpelvl.2 |
Ref | Expression |
---|---|
genpmu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 6573 | . . . 4 | |
2 | prmu 6576 | . . . 4 | |
3 | rexex 2368 | . . . 4 | |
4 | 1, 2, 3 | 3syl 17 | . . 3 |
5 | 4 | adantr 261 | . 2 |
6 | prop 6573 | . . . . 5 | |
7 | prmu 6576 | . . . . 5 | |
8 | rexex 2368 | . . . . 5 | |
9 | 6, 7, 8 | 3syl 17 | . . . 4 |
10 | 9 | ad2antlr 458 | . . 3 |
11 | genpelvl.1 | . . . . . . 7 | |
12 | genpelvl.2 | . . . . . . 7 | |
13 | 11, 12 | genppreclu 6613 | . . . . . 6 |
14 | 13 | imp 115 | . . . . 5 |
15 | elprnqu 6580 | . . . . . . . . . 10 | |
16 | 1, 15 | sylan 267 | . . . . . . . . 9 |
17 | elprnqu 6580 | . . . . . . . . . 10 | |
18 | 6, 17 | sylan 267 | . . . . . . . . 9 |
19 | 16, 18 | anim12i 321 | . . . . . . . 8 |
20 | 19 | an4s 522 | . . . . . . 7 |
21 | 12 | caovcl 5655 | . . . . . . 7 |
22 | 20, 21 | syl 14 | . . . . . 6 |
23 | simpr 103 | . . . . . . 7 | |
24 | 23 | eleq1d 2106 | . . . . . 6 |
25 | 22, 24 | rspcedv 2660 | . . . . 5 |
26 | 14, 25 | mpd 13 | . . . 4 |
27 | 26 | anassrs 380 | . . 3 |
28 | 10, 27 | exlimddv 1778 | . 2 |
29 | 5, 28 | exlimddv 1778 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wceq 1243 wex 1381 wcel 1393 wrex 2307 crab 2310 cop 3378 cfv 4902 (class class class)co 5512 cmpt2 5514 c1st 5765 c2nd 5766 cnq 6378 cnp 6389 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-qs 6112 df-ni 6402 df-nqqs 6446 df-inp 6564 |
This theorem is referenced by: addclpr 6635 mulclpr 6670 |
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