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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 9403 | . . 3 | |
2 | rpre 9403 | . . 3 | |
3 | remulcl 7716 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | elrp 9399 | . . 3 | |
6 | elrp 9399 | . . 3 | |
7 | mulgt0 7807 | . . 3 | |
8 | 5, 6, 7 | syl2anb 289 | . 2 |
9 | elrp 9399 | . 2 | |
10 | 4, 8, 9 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 cmul 7593 clt 7768 crp 9397 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-mulrcl 7687 ax-rnegex 7697 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-rp 9398 |
This theorem is referenced by: rpmulcld 9455 rpexpcl 10267 expcnvap0 11226 |
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