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Theorem mulcanapd 8335
Description: Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.)
Hypotheses
Ref Expression
mulcand.1  |-  ( ph  ->  A  e.  CC )
mulcand.2  |-  ( ph  ->  B  e.  CC )
mulcand.3  |-  ( ph  ->  C  e.  CC )
mulcand.4  |-  ( ph  ->  C #  0 )
Assertion
Ref Expression
mulcanapd  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  <-> 
A  =  B ) )

Proof of Theorem mulcanapd
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mulcand.3 . . . 4  |-  ( ph  ->  C  e.  CC )
2 mulcand.4 . . . 4  |-  ( ph  ->  C #  0 )
3 recexap 8327 . . . 4  |-  ( ( C  e.  CC  /\  C #  0 )  ->  E. x  e.  CC  ( C  x.  x )  =  1 )
41, 2, 3syl2anc 406 . . 3  |-  ( ph  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
5 oveq2 5736 . . . 4  |-  ( ( C  x.  A )  =  ( C  x.  B )  ->  (
x  x.  ( C  x.  A ) )  =  ( x  x.  ( C  x.  B
) ) )
6 simprl 503 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  x  e.  CC )
71adantr 272 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  C  e.  CC )
86, 7mulcomd 7711 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  ( C  x.  x ) )
9 simprr 504 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( C  x.  x
)  =  1 )
108, 9eqtrd 2147 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  1 )
1110oveq1d 5743 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( 1  x.  A ) )
12 mulcand.1 . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
1312adantr 272 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  A  e.  CC )
146, 7, 13mulassd 7713 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( x  x.  ( C  x.  A ) ) )
1513mulid2d 7708 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
1611, 14, 153eqtr3d 2155 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  A )
)  =  A )
1710oveq1d 5743 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( 1  x.  B ) )
18 mulcand.2 . . . . . . . 8  |-  ( ph  ->  B  e.  CC )
1918adantr 272 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  B  e.  CC )
206, 7, 19mulassd 7713 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( x  x.  ( C  x.  B ) ) )
2119mulid2d 7708 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  B
)  =  B )
2217, 20, 213eqtr3d 2155 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  B )
)  =  B )
2316, 22eqeq12d 2129 . . . 4  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  ( C  x.  A
) )  =  ( x  x.  ( C  x.  B ) )  <-> 
A  =  B ) )
245, 23syl5ib 153 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
254, 24rexlimddv 2528 . 2  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
26 oveq2 5736 . 2  |-  ( A  =  B  ->  ( C  x.  A )  =  ( C  x.  B ) )
2725, 26impbid1 141 1  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2391   class class class wbr 3895  (class class class)co 5728   CCcc 7545   0cc0 7547   1c1 7548    x. cmul 7552   # cap 8261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262
This theorem is referenced by:  mulcanap2d  8336  mulcanapad  8337  mulcanap  8339  div11ap  8373  eqneg  8405  dvdscmulr  11370  qredeq  11623  cncongr1  11630
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