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Theorem xlemul1 11253
Description: Extended real version of lemul1 10181. (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 10998 . . . 4  |-  ( C  e.  RR+  ->  C  e. 
RR* )
2 rpge0 11003 . . . 4  |-  ( C  e.  RR+  ->  0  <_  C )
31, 2jca 532 . . 3  |-  ( C  e.  RR+  ->  ( C  e.  RR*  /\  0  <_  C ) )
4 xlemul1a 11251 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR*  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( A xe C )  <_  ( B xe C ) )
54ex 434 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR*  /\  0  <_  C ) )  -> 
( A  <_  B  ->  ( A xe C )  <_  ( B xe C ) ) )
63, 5syl3an3 1253 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <_  B  ->  ( A xe C )  <_  ( B xe C ) ) )
7 simp1 988 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  A  e. 
RR* )
813ad2ant3 1011 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  e. 
RR* )
9 xmulcl 11236 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A xe C )  e.  RR* )
107, 8, 9syl2anc 661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A xe C )  e.  RR* )
11 simp2 989 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  B  e. 
RR* )
12 xmulcl 11236 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B xe C )  e.  RR* )
1311, 8, 12syl2anc 661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( B xe C )  e.  RR* )
14 rpreccl 11014 . . . . . 6  |-  ( C  e.  RR+  ->  ( 1  /  C )  e.  RR+ )
15143ad2ant3 1011 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( 1  /  C )  e.  RR+ )
16 rpxr 10998 . . . . 5  |-  ( ( 1  /  C )  e.  RR+  ->  ( 1  /  C )  e. 
RR* )
1715, 16syl 16 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( 1  /  C )  e. 
RR* )
18 rpge0 11003 . . . . 5  |-  ( ( 1  /  C )  e.  RR+  ->  0  <_ 
( 1  /  C
) )
1915, 18syl 16 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  0  <_ 
( 1  /  C
) )
20 xlemul1a 11251 . . . . 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 434 . . . 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 1222 . . 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 11250 . . . . . 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 1218 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( A xe C ) xe ( 1  /  C ) )  =  ( A xe ( C xe ( 1  /  C ) ) ) )
25 rpre 10997 . . . . . . . . 9  |-  ( C  e.  RR+  ->  C  e.  RR )
26253ad2ant3 1011 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  e.  RR )
2715rpred 11027 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( 1  /  C )  e.  RR )
28 rexmul 11234 . . . . . . . 8  |-  ( ( C  e.  RR  /\  ( 1  /  C
)  e.  RR )  ->  ( C xe ( 1  /  C ) )  =  ( C  x.  (
1  /  C ) ) )
2926, 27, 28syl2anc 661 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( C xe ( 1  /  C ) )  =  ( C  x.  ( 1  /  C
) ) )
3026recnd 9412 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  e.  CC )
31 rpne0 11006 . . . . . . . . 9  |-  ( C  e.  RR+  ->  C  =/=  0 )
32313ad2ant3 1011 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  =/=  0 )
3330, 32recidd 10102 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( C  x.  ( 1  /  C ) )  =  1 )
3429, 33eqtrd 2475 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( C xe ( 1  /  C ) )  =  1 )
3534oveq2d 6107 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A xe ( C xe ( 1  /  C ) ) )  =  ( A xe 1 ) )
36 xmulid1 11242 . . . . . 6  |-  ( A  e.  RR*  ->  ( A xe 1 )  =  A )
377, 36syl 16 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A xe 1 )  =  A )
3824, 35, 373eqtrd 2479 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( A xe C ) xe ( 1  /  C ) )  =  A )
39 xmulass 11250 . . . . . 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 1218 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( B xe C ) xe ( 1  /  C ) )  =  ( B xe ( C xe ( 1  /  C ) ) ) )
4134oveq2d 6107 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( B xe ( C xe ( 1  /  C ) ) )  =  ( B xe 1 ) )
42 xmulid1 11242 . . . . . 6  |-  ( B  e.  RR*  ->  ( B xe 1 )  =  B )
4311, 42syl 16 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( B xe 1 )  =  B )
4440, 41, 433eqtrd 2479 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( B xe C ) xe ( 1  /  C ) )  =  B )
4538, 44breq12d 4305 . . 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 214 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( A xe C )  <_  ( B xe C )  ->  A  <_  B
) )
476, 46impbid 191 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   class class class wbr 4292  (class class class)co 6091   RRcr 9281   0cc0 9282   1c1 9283    x. cmul 9287   RR*cxr 9417    <_ cle 9419    / cdiv 9993   RR+crp 10991   xecxmu 11088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-rp 10992  df-xneg 11089  df-xmul 11091
This theorem is referenced by:  xlemul2  11254  xltmul1  11255  nmoleub2lem  20669  xrmulc1cn  26360
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