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Theorem xlemul1 11494
Description: Extended real version of lemul1 10406. (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 11239 . . . 4  |-  ( C  e.  RR+  ->  C  e. 
RR* )
2 rpge0 11244 . . . 4  |-  ( C  e.  RR+  ->  0  <_  C )
31, 2jca 532 . . 3  |-  ( C  e.  RR+  ->  ( C  e.  RR*  /\  0  <_  C ) )
4 xlemul1a 11492 . . . 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 1263 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A  <_  B  ->  ( A xe C )  <_  ( B xe C ) ) )
7 simp1 996 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  A  e. 
RR* )
813ad2ant3 1019 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  e. 
RR* )
9 xmulcl 11477 . . . . 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 997 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  B  e. 
RR* )
12 xmulcl 11477 . . . . 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 11255 . . . . . 6  |-  ( C  e.  RR+  ->  ( 1  /  C )  e.  RR+ )
15143ad2ant3 1019 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( 1  /  C )  e.  RR+ )
16 rpxr 11239 . . . . 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 11244 . . . . 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 11492 . . . . 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 1232 . . 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 11491 . . . . . 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 1228 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( A xe C ) xe ( 1  /  C ) )  =  ( A xe ( C xe ( 1  /  C ) ) ) )
25 rpre 11238 . . . . . . . . 9  |-  ( C  e.  RR+  ->  C  e.  RR )
26253ad2ant3 1019 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  e.  RR )
2715rpred 11268 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( 1  /  C )  e.  RR )
28 rexmul 11475 . . . . . . . 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 9634 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  e.  CC )
31 rpne0 11247 . . . . . . . . 9  |-  ( C  e.  RR+  ->  C  =/=  0 )
32313ad2ant3 1019 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  C  =/=  0 )
3330, 32recidd 10327 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( C  x.  ( 1  /  C ) )  =  1 )
3429, 33eqtrd 2508 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( C xe ( 1  /  C ) )  =  1 )
3534oveq2d 6311 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( A xe ( C xe ( 1  /  C ) ) )  =  ( A xe 1 ) )
36 xmulid1 11483 . . . . . 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 2512 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( A xe C ) xe ( 1  /  C ) )  =  A )
39 xmulass 11491 . . . . . 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 1228 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( B xe C ) xe ( 1  /  C ) )  =  ( B xe ( C xe ( 1  /  C ) ) ) )
4134oveq2d 6311 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( B xe ( C xe ( 1  /  C ) ) )  =  ( B xe 1 ) )
42 xmulid1 11483 . . . . . 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 2512 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR+ )  ->  ( ( B xe C ) xe ( 1  /  C ) )  =  B )
4538, 44breq12d 4466 . . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505    x. cmul 9509   RR*cxr 9639    <_ cle 9641    / cdiv 10218   RR+crp 11232   xecxmu 11329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-rp 11233  df-xneg 11330  df-xmul 11332
This theorem is referenced by:  xlemul2  11495  xltmul1  11496  nmoleub2lem  21465  xrmulc1cn  27737
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