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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 7730 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axmulgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-mulgt0 7730 | . 2 | |
2 | 0re 7759 | . . . 4 | |
3 | ltxrlt 7823 | . . . 4 | |
4 | 2, 3 | mpan 420 | . . 3 |
5 | ltxrlt 7823 | . . . 4 | |
6 | 2, 5 | mpan 420 | . . 3 |
7 | 4, 6 | bi2anan9 595 | . 2 |
8 | remulcl 7741 | . . 3 | |
9 | ltxrlt 7823 | . . 3 | |
10 | 2, 8, 9 | sylancr 410 | . 2 |
11 | 1, 7, 10 | 3imtr4d 202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 cc0 7613 cltrr 7617 cmul 7618 clt 7793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-mulrcl 7712 ax-rnegex 7722 ax-pre-mulgt0 7730 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-pnf 7795 df-mnf 7796 df-ltxr 7798 |
This theorem is referenced by: mulgt0 7832 mulgt0i 7866 sin02gt0 11459 sinq12gt0 12900 |
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