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Theorem xmulcand 27449
Description: Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
Hypotheses
Ref Expression
xmulcand.1  |-  ( ph  ->  A  e.  RR* )
xmulcand.2  |-  ( ph  ->  B  e.  RR* )
xmulcand.3  |-  ( ph  ->  C  e.  RR )
xmulcand.4  |-  ( ph  ->  C  =/=  0 )
Assertion
Ref Expression
xmulcand  |-  ( ph  ->  ( ( C xe A )  =  ( C xe B )  <->  A  =  B ) )

Proof of Theorem xmulcand
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 xmulcand.3 . . . 4  |-  ( ph  ->  C  e.  RR )
2 xmulcand.4 . . . 4  |-  ( ph  ->  C  =/=  0 )
3 xrecex 27448 . . . 4  |-  ( ( C  e.  RR  /\  C  =/=  0 )  ->  E. x  e.  RR  ( C xe x )  =  1 )
41, 2, 3syl2anc 661 . . 3  |-  ( ph  ->  E. x  e.  RR  ( C xe x )  =  1 )
5 oveq2 6303 . . . 4  |-  ( ( C xe A )  =  ( C xe B )  ->  ( x xe ( C xe A ) )  =  ( x xe ( C xe B ) ) )
6 simprl 755 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  x  e.  RR )
76rexrd 9655 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  x  e.  RR* )
81adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  C  e.  RR )
98rexrd 9655 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  C  e.  RR* )
10 xmulcom 11470 . . . . . . . . 9  |-  ( ( x  e.  RR*  /\  C  e.  RR* )  ->  (
x xe C )  =  ( C xe x ) )
117, 9, 10syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe C )  =  ( C xe x ) )
12 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( C xe x )  =  1 )
1311, 12eqtrd 2508 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe C )  =  1 )
1413oveq1d 6310 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe C ) xe A )  =  ( 1 xe A ) )
15 xmulcand.1 . . . . . . . 8  |-  ( ph  ->  A  e.  RR* )
1615adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  A  e.  RR* )
17 xmulass 11491 . . . . . . 7  |-  ( ( x  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  ->  (
( x xe C ) xe A )  =  ( x xe ( C xe A ) ) )
187, 9, 16, 17syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe C ) xe A )  =  ( x xe ( C xe A ) ) )
19 xmulid2 11484 . . . . . . 7  |-  ( A  e.  RR*  ->  ( 1 xe A )  =  A )
2016, 19syl 16 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( 1 xe A )  =  A )
2114, 18, 203eqtr3d 2516 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe ( C xe A ) )  =  A )
2213oveq1d 6310 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe C ) xe B )  =  ( 1 xe B ) )
23 xmulcand.2 . . . . . . . 8  |-  ( ph  ->  B  e.  RR* )
2423adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  B  e.  RR* )
25 xmulass 11491 . . . . . . 7  |-  ( ( x  e.  RR*  /\  C  e.  RR*  /\  B  e. 
RR* )  ->  (
( x xe C ) xe B )  =  ( x xe ( C xe B ) ) )
267, 9, 24, 25syl3anc 1228 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe C ) xe B )  =  ( x xe ( C xe B ) ) )
27 xmulid2 11484 . . . . . . 7  |-  ( B  e.  RR*  ->  ( 1 xe B )  =  B )
2824, 27syl 16 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( 1 xe B )  =  B )
2922, 26, 283eqtr3d 2516 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe ( C xe B ) )  =  B )
3021, 29eqeq12d 2489 . . . 4  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe ( C xe A ) )  =  ( x xe ( C xe B ) )  <->  A  =  B
) )
315, 30syl5ib 219 . . 3  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( C xe A )  =  ( C xe B )  ->  A  =  B )
)
324, 31rexlimddv 2963 . 2  |-  ( ph  ->  ( ( C xe A )  =  ( C xe B )  ->  A  =  B ) )
33 oveq2 6303 . 2  |-  ( A  =  B  ->  ( C xe A )  =  ( C xe B ) )
3432, 33impbid1 203 1  |-  ( ph  ->  ( ( C xe A )  =  ( C xe B )  <->  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818  (class class class)co 6295   RRcr 9503   0cc0 9504   1c1 9505   RR*cxr 9639   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-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-xneg 11330  df-xmul 11332
This theorem is referenced by:  xreceu  27450
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