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Mirrors > Home > ILE Home > Th. List > rpgecl | Unicode version |
Description: A number greater or equal to a positive real is positive real. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpgecl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 939 | . 2 | |
2 | 0red 7120 | . . 3 | |
3 | rpre 8740 | . . . 4 | |
4 | 3 | 3ad2ant1 959 | . . 3 |
5 | rpgt0 8745 | . . . 4 | |
6 | 5 | 3ad2ant1 959 | . . 3 |
7 | simp3 940 | . . 3 | |
8 | 2, 4, 1, 6, 7 | ltletrd 7527 | . 2 |
9 | elrp 8736 | . 2 | |
10 | 1, 8, 9 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 919 wcel 1433 class class class wbr 3785 cr 6980 cc0 6981 clt 7153 cle 7154 crp 8734 |
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-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addrcl 7073 ax-rnegex 7085 ax-pre-ltwlin 7089 |
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-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 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-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-rp 8735 |
This theorem is referenced by: divge1 8800 rpgecld 8813 |
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