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Theorem mulcanapd 7807
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 7799 . . . 4  |-  ( ( C  e.  CC  /\  C #  0 )  ->  E. x  e.  CC  ( C  x.  x )  =  1 )
41, 2, 3syl2anc 403 . . 3  |-  ( ph  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
5 oveq2 5545 . . . 4  |-  ( ( C  x.  A )  =  ( C  x.  B )  ->  (
x  x.  ( C  x.  A ) )  =  ( x  x.  ( C  x.  B
) ) )
6 simprl 498 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  x  e.  CC )
71adantr 270 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  C  e.  CC )
86, 7mulcomd 7191 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  ( C  x.  x ) )
9 simprr 499 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( C  x.  x
)  =  1 )
108, 9eqtrd 2114 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  1 )
1110oveq1d 5552 . . . . . 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 270 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  A  e.  CC )
146, 7, 13mulassd 7193 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( x  x.  ( C  x.  A ) ) )
1513mulid2d 7188 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
1611, 14, 153eqtr3d 2122 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  A )
)  =  A )
1710oveq1d 5552 . . . . . 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 270 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  B  e.  CC )
206, 7, 19mulassd 7193 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( x  x.  ( C  x.  B ) ) )
2119mulid2d 7188 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  B
)  =  B )
2217, 20, 213eqtr3d 2122 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  B )
)  =  B )
2316, 22eqeq12d 2096 . . . 4  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  ( C  x.  A
) )  =  ( x  x.  ( C  x.  B ) )  <-> 
A  =  B ) )
245, 23syl5ib 152 . . 3  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
254, 24rexlimddv 2482 . 2  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
26 oveq2 5545 . 2  |-  ( A  =  B  ->  ( C  x.  A )  =  ( C  x.  B ) )
2725, 26impbid1 140 1  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   E.wrex 2350   class class class wbr 3787  (class class class)co 5537   CCcc 7030   0cc0 7032   1c1 7033    x. cmul 7037   # cap 7737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-po 4053  df-iso 4054  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738
This theorem is referenced by:  mulcanap2d  7808  mulcanapad  7809  mulcanap  7811  div11ap  7844  eqneg  7876  dvdscmulr  10358  qredeq  10611  cncongr1  10618
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