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Mirrors > Home > ILE Home > Th. List > rpmulcld | Unicode version |
Description: Closure law for multiplication of positive reals. Part of Axiom 7 of [Apostol] p. 20. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 |
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rpaddcld.1 |
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Ref | Expression |
---|---|
rpmulcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 |
. 2
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2 | rpaddcld.1 |
. 2
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3 | rpmulcl 9495 |
. 2
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4 | 1, 2, 3 | syl2anc 409 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1re 7738 ax-addrcl 7741 ax-mulrcl 7743 ax-rnegex 7753 ax-pre-mulgt0 7761 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-rp 9471 |
This theorem is referenced by: qbtwnrelemcalc 10064 cvg1nlemcxze 10786 cvg1nlemres 10789 resqrexlemnm 10822 resqrexlemcvg 10823 reccn2ap 11114 cvgratnnlembern 11324 cvgratnnlemrate 11331 cvgratnn 11332 eirraplem 11519 cosordlem 12978 rpmulcxp 13038 |
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