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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 7930 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
axmulgt0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-pre-mulgt0 7930 |
. 2
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2 | 0re 7959 |
. . . 4
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3 | ltxrlt 8025 |
. . . 4
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4 | 2, 3 | mpan 424 |
. . 3
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5 | ltxrlt 8025 |
. . . 4
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6 | 2, 5 | mpan 424 |
. . 3
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7 | 4, 6 | bi2anan9 606 |
. 2
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8 | remulcl 7941 |
. . 3
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9 | ltxrlt 8025 |
. . 3
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10 | 2, 8, 9 | sylancr 414 |
. 2
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11 | 1, 7, 10 | 3imtr4d 203 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1re 7907 ax-addrcl 7910 ax-mulrcl 7912 ax-rnegex 7922 ax-pre-mulgt0 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-pnf 7996 df-mnf 7997 df-ltxr 7999 |
This theorem is referenced by: mulgt0 8034 mulgt0i 8069 sin02gt0 11773 sinq12gt0 14336 |
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