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Mirrors > Home > ILE Home > Th. List > rpmulcl | Unicode version |
Description: Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by NM, 27-Oct-2007.) |
Ref | Expression |
---|---|
rpmulcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 9658 |
. . 3
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2 | rpre 9658 |
. . 3
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3 | remulcl 7938 |
. . 3
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4 | 1, 2, 3 | syl2an 289 |
. 2
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5 | elrp 9653 |
. . 3
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6 | elrp 9653 |
. . 3
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7 | mulgt0 8030 |
. . 3
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8 | 5, 6, 7 | syl2anb 291 |
. 2
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9 | elrp 9653 |
. 2
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10 | 4, 8, 9 | sylanbrc 417 |
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 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 ax-mulrcl 7909 ax-rnegex 7919 ax-pre-mulgt0 7927 |
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 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-xp 4632 df-pnf 7992 df-mnf 7993 df-ltxr 7995 df-rp 9652 |
This theorem is referenced by: rpmulcld 9711 rpexpcl 10536 expcnvap0 11505 fprodrpcl 11614 cosordlem 14163 rprelogbmul 14266 taupi 14702 |
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