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Theorem xlemul1 12120
Description: Extended real version of lemul1 10875. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xlemul1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A <_ B <-> ( A xe C ) <_ ( B xe C ) ) )
Proof of Theorem xlemul1
StepHypRef Expression
1 rpxr 11840 . . . 4 |- ( C e. RR+ -> C e. RR* )
2 rpge0 11845 . . . 4 |- ( C e. RR+ -> 0 <_ C )
31, 2jca 554 . . 3 |- ( C e. RR+ -> ( C e. RR* /\ 0 <_ C ) )
4 xlemul1a 12118 . . . 4 |- ( ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) /\ A <_ B ) -> ( A xe C ) <_ ( B xe C ) )
54ex 450 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ ( C e. RR* /\ 0 <_ C ) ) -> ( A <_ B -> ( A xe C ) <_ ( B xe C ) ) )
63, 5syl3an3 1361 . 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A <_ B -> ( A xe C ) <_ ( B xe C ) ) )
7 simp1 1061 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> A e. RR* )
813ad2ant3 1084 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> C e. RR* )
9 xmulcl 12103 . . . . 5 |- ( ( A e. RR* /\ C e. RR* ) -> ( A xe C ) e. RR* )
107, 8, 9syl2anc 693 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A xe C ) e. RR* )
11 simp2 1062 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> B e. RR* )
12 xmulcl 12103 . . . . 5 |- ( ( B e. RR* /\ C e. RR* ) -> ( B xe C ) e. RR* )
1311, 8, 12syl2anc 693 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( B xe C ) e. RR* )
14 rpreccl 11857 . . . . . 6 |- ( C e. RR+ -> ( 1 / C ) e. RR+ )
15143ad2ant3 1084 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( 1 / C ) e. RR+ )
16 rpxr 11840 . . . . 5 |- ( ( 1 / C ) e. RR+ -> ( 1 / C ) e. RR* )
1715, 16syl 17 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( 1 / C ) e. RR* )
18 rpge0 11845 . . . . 5 |- ( ( 1 / C ) e. RR+ -> 0 <_ ( 1 / C ) )
1915, 18syl 17 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> 0 <_ ( 1 / C ) )
20 xlemul1a 12118 . . . . 5 |- ( ( ( ( A xe C ) e. RR* /\ ( B xe C ) e. RR* /\ ( ( 1 / C ) e. RR* /\ 0 <_ ( 1 / C ) ) ) /\ ( A xe C ) <_ ( B xe C ) ) -> ( ( A xe C ) xe ( 1 / C ) ) <_ ( ( B xe C ) xe ( 1 / C ) ) )
2120ex 450 . . . 4 |- ( ( ( A xe C ) e. RR* /\ ( B xe C ) e. RR* /\ ( ( 1 / C ) e. RR* /\ 0 <_ ( 1 / C ) ) ) -> ( ( A xe C ) <_ ( B xe C ) -> ( ( A xe C ) xe ( 1 / C ) ) <_ ( ( B xe C ) xe ( 1 / C ) ) ) )
2210, 13, 17, 19, 21syl112anc 1330 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A xe C ) <_ ( B xe C ) -> ( ( A xe C ) xe ( 1 / C ) ) <_ ( ( B xe C ) xe ( 1 / C ) ) ) )
23 xmulass 12117 . . . . . 6 |- ( ( A e. RR* /\ C e. RR* /\ ( 1 / C ) e. RR* ) -> ( ( A xe C ) xe ( 1 / C ) ) = ( A xe ( C xe ( 1 / C ) ) ) )
247, 8, 17, 23syl3anc 1326 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A xe C ) xe ( 1 / C ) ) = ( A xe ( C xe ( 1 / C ) ) ) )
25 rpre 11839 . . . . . . . . 9 |- ( C e. RR+ -> C e. RR )
26253ad2ant3 1084 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> C e. RR )
2715rpred 11872 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( 1 / C ) e. RR )
28 rexmul 12101 . . . . . . . 8 |- ( ( C e. RR /\ ( 1 / C ) e. RR ) -> ( C xe ( 1 / C ) ) = ( C x. ( 1 / C ) ) )
2926, 27, 28syl2anc 693 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( C xe ( 1 / C ) ) = ( C x. ( 1 / C ) ) )
3026recnd 10068 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> C e. CC )
31 rpne0 11848 . . . . . . . . 9 |- ( C e. RR+ -> C =/= 0 )
32313ad2ant3 1084 . . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> C =/= 0 )
3330, 32recidd 10796 . . . . . . 7 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( C x. ( 1 / C ) ) = 1 )
3429, 33eqtrd 2656 . . . . . 6 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( C xe ( 1 / C ) ) = 1 )
3534oveq2d 6666 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A xe ( C xe ( 1 / C ) ) ) = ( A xe 1 ) )
36 xmulid1 12109 . . . . . 6 |- ( A e. RR* -> ( A xe 1 ) = A )
377, 36syl 17 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A xe 1 ) = A )
3824, 35, 373eqtrd 2660 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A xe C ) xe ( 1 / C ) ) = A )
39 xmulass 12117 . . . . . 6 |- ( ( B e. RR* /\ C e. RR* /\ ( 1 / C ) e. RR* ) -> ( ( B xe C ) xe ( 1 / C ) ) = ( B xe ( C xe ( 1 / C ) ) ) )
4011, 8, 17, 39syl3anc 1326 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( B xe C ) xe ( 1 / C ) ) = ( B xe ( C xe ( 1 / C ) ) ) )
4134oveq2d 6666 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( B xe ( C xe ( 1 / C ) ) ) = ( B xe 1 ) )
42 xmulid1 12109 . . . . . 6 |- ( B e. RR* -> ( B xe 1 ) = B )
4311, 42syl 17 . . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( B xe 1 ) = B )
4440, 41, 433eqtrd 2660 . . . 4 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( B xe C ) xe ( 1 / C ) ) = B )
4538, 44breq12d 4666 . . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( ( A xe C ) xe ( 1 / C ) ) <_ ( ( B xe C ) xe ( 1 / C ) ) <-> A <_ B ) )
4622, 45sylibd 229 . 2 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( ( A xe C ) <_ ( B xe C ) -> A <_ B ) )
476, 46impbid 202 1 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR+ ) -> ( A <_ B <-> ( A xe C ) <_ ( B xe C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   RR*cxr 10073   <_ cle 10075   / cdiv 10684   RR+crp 11832   xecxmu 11945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-rp 11833  df-xneg 11946  df-xmul 11948
This theorem is referenced by:  xlemul2  12121  xltmul1  12122  nmoleub2lem  22914  xrmulc1cn  29976
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