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Theorem xmulcand 26101
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 26100 . . . 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 6104 . . . 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 9438 . . . . . . . . 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 9438 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  C  e.  RR* )
10 xmulcom 11234 . . . . . . . . 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 2475 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe C )  =  1 )
1413oveq1d 6111 . . . . . 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 11255 . . . . . . 7  |-  ( ( x  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  ->  (
( x xe C ) xe A )  =  ( x xe ( C xe A ) ) )
187, 9, 16, 17syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe C ) xe A )  =  ( x xe ( C xe A ) ) )
19 xmulid2 11248 . . . . . . 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 2483 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe ( C xe A ) )  =  A )
2213oveq1d 6111 . . . . . 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 11255 . . . . . . 7  |-  ( ( x  e.  RR*  /\  C  e.  RR*  /\  B  e. 
RR* )  ->  (
( x xe C ) xe B )  =  ( x xe ( C xe B ) ) )
267, 9, 24, 25syl3anc 1218 . . . . . 6  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( ( x xe C ) xe B )  =  ( x xe ( C xe B ) ) )
27 xmulid2 11248 . . . . . . 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 2483 . . . . 5  |-  ( (
ph  /\  ( x  e.  RR  /\  ( C xe x )  =  1 ) )  ->  ( x xe ( C xe B ) )  =  B )
3021, 29eqeq12d 2457 . . . 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 2850 . 2  |-  ( ph  ->  ( ( C xe A )  =  ( C xe B )  ->  A  =  B ) )
33 oveq2 6104 . 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 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721  (class class class)co 6096   RRcr 9286   0cc0 9287   1c1 9288   RR*cxr 9422   xecxmu 11093
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-xneg 11094  df-xmul 11096
This theorem is referenced by:  xreceu  26102
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