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Mirrors > Home > ILE Home > Th. List > genpml | Unicode version |
Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) |
Ref | Expression |
---|---|
genpelvl.1 | |
genpelvl.2 |
Ref | Expression |
---|---|
genpml |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 6665 | . . . 4 | |
2 | prml 6667 | . . . 4 | |
3 | rexex 2410 | . . . 4 | |
4 | 1, 2, 3 | 3syl 17 | . . 3 |
5 | 4 | adantr 270 | . 2 |
6 | prop 6665 | . . . . 5 | |
7 | prml 6667 | . . . . 5 | |
8 | rexex 2410 | . . . . 5 | |
9 | 6, 7, 8 | 3syl 17 | . . . 4 |
10 | 9 | ad2antlr 472 | . . 3 |
11 | genpelvl.1 | . . . . . . 7 | |
12 | genpelvl.2 | . . . . . . 7 | |
13 | 11, 12 | genpprecll 6704 | . . . . . 6 |
14 | 13 | imp 122 | . . . . 5 |
15 | elprnql 6671 | . . . . . . . . . 10 | |
16 | 1, 15 | sylan 277 | . . . . . . . . 9 |
17 | elprnql 6671 | . . . . . . . . . 10 | |
18 | 6, 17 | sylan 277 | . . . . . . . . 9 |
19 | 16, 18 | anim12i 331 | . . . . . . . 8 |
20 | 19 | an4s 552 | . . . . . . 7 |
21 | 12 | caovcl 5675 | . . . . . . 7 |
22 | 20, 21 | syl 14 | . . . . . 6 |
23 | simpr 108 | . . . . . . 7 | |
24 | 23 | eleq1d 2147 | . . . . . 6 |
25 | 22, 24 | rspcedv 2705 | . . . . 5 |
26 | 14, 25 | mpd 13 | . . . 4 |
27 | 26 | anassrs 392 | . . 3 |
28 | 10, 27 | exlimddv 1819 | . 2 |
29 | 5, 28 | exlimddv 1819 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wceq 1284 wex 1421 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: addclpr 6727 mulclpr 6762 |
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