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Mirrors > Home > ILE Home > Th. List > axmulgt0 | Unicode version |
Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7861 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axmulgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-mulgt0 7861 | . 2 | |
2 | 0re 7890 | . . . 4 | |
3 | ltxrlt 7955 | . . . 4 | |
4 | 2, 3 | mpan 421 | . . 3 |
5 | ltxrlt 7955 | . . . 4 | |
6 | 2, 5 | mpan 421 | . . 3 |
7 | 4, 6 | bi2anan9 596 | . 2 |
8 | remulcl 7872 | . . 3 | |
9 | ltxrlt 7955 | . . 3 | |
10 | 2, 8, 9 | sylancr 411 | . 2 |
11 | 1, 7, 10 | 3imtr4d 202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2135 class class class wbr 3976 (class class class)co 5836 cr 7743 cc0 7744 cltrr 7748 cmul 7749 clt 7924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1re 7838 ax-addrcl 7841 ax-mulrcl 7843 ax-rnegex 7853 ax-pre-mulgt0 7861 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-pnf 7926 df-mnf 7927 df-ltxr 7929 |
This theorem is referenced by: mulgt0 7964 mulgt0i 7999 sin02gt0 11690 sinq12gt0 13298 |
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