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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 9562 | . . 3 | |
2 | rpre 9562 | . . 3 | |
3 | remulcl 7855 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | elrp 9557 | . . 3 | |
6 | elrp 9557 | . . 3 | |
7 | mulgt0 7947 | . . 3 | |
8 | 5, 6, 7 | syl2anb 289 | . 2 |
9 | elrp 9557 | . 2 | |
10 | 4, 8, 9 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 class class class wbr 3965 (class class class)co 5821 cr 7726 cc0 7727 cmul 7732 clt 7907 crp 9555 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1re 7821 ax-addrcl 7824 ax-mulrcl 7826 ax-rnegex 7836 ax-pre-mulgt0 7844 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-xp 4591 df-pnf 7909 df-mnf 7910 df-ltxr 7912 df-rp 9556 |
This theorem is referenced by: rpmulcld 9615 rpexpcl 10433 expcnvap0 11394 fprodrpcl 11503 cosordlem 13157 rprelogbmul 13259 taupi 13628 |
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