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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 7656 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 7656 |
. 2
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2 | 0re 7684 |
. . . 4
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3 | ltxrlt 7748 |
. . . 4
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4 | 2, 3 | mpan 418 |
. . 3
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5 | ltxrlt 7748 |
. . . 4
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6 | 2, 5 | mpan 418 |
. . 3
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7 | 4, 6 | bi2anan9 578 |
. 2
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8 | remulcl 7666 |
. . 3
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9 | ltxrlt 7748 |
. . 3
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10 | 2, 8, 9 | sylancr 408 |
. 2
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11 | 1, 7, 10 | 3imtr4d 202 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-1re 7633 ax-addrcl 7636 ax-mulrcl 7638 ax-rnegex 7648 ax-pre-mulgt0 7656 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-xp 4503 df-pnf 7720 df-mnf 7721 df-ltxr 7723 |
This theorem is referenced by: mulgt0 7756 mulgt0i 7790 sin02gt0 11315 |
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