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Theorem mulcand 9417
Description: Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18. (Contributed by NM, 26-Jan-1995.) (Revised by Mario Carneiro, 27-May-2016.)
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
mulcand  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  <-> 
A  =  B ) )

Proof of Theorem mulcand
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 recex 9416 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
41, 2, 3syl2anc 642 . . 3  |-  ( ph  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
5 oveq2 5882 . . . . . 6  |-  ( ( C  x.  A )  =  ( C  x.  B )  ->  (
x  x.  ( C  x.  A ) )  =  ( x  x.  ( C  x.  B
) ) )
6 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  x  e.  CC )
71adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  C  e.  CC )
86, 7mulcomd 8872 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  ( C  x.  x ) )
9 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( C  x.  x
)  =  1 )
108, 9eqtrd 2328 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  1 )
1110oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( 1  x.  A ) )
12 mulcand.1 . . . . . . . . . 10  |-  ( ph  ->  A  e.  CC )
1312adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  A  e.  CC )
146, 7, 13mulassd 8874 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( x  x.  ( C  x.  A ) ) )
1513mulid2d 8869 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
1611, 14, 153eqtr3d 2336 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  A )
)  =  A )
1710oveq1d 5889 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( 1  x.  B ) )
18 mulcand.2 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
1918adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  B  e.  CC )
206, 7, 19mulassd 8874 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( x  x.  ( C  x.  B ) ) )
2119mulid2d 8869 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  B
)  =  B )
2217, 20, 213eqtr3d 2336 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  B )
)  =  B )
2316, 22eqeq12d 2310 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  ( C  x.  A
) )  =  ( x  x.  ( C  x.  B ) )  <-> 
A  =  B ) )
245, 23syl5ib 210 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
2524expr 598 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( C  x.  x )  =  1  ->  (
( C  x.  A
)  =  ( C  x.  B )  ->  A  =  B )
) )
2625rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. x  e.  CC  ( C  x.  x )  =  1  ->  ( ( C  x.  A )  =  ( C  x.  B
)  ->  A  =  B ) ) )
274, 26mpd 14 . 2  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
28 oveq2 5882 . 2  |-  ( A  =  B  ->  ( C  x.  A )  =  ( C  x.  B ) )
2927, 28impbid1 194 1  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557  (class class class)co 5874   CCcc 8751   0cc0 8753   1c1 8754    x. cmul 8758
This theorem is referenced by:  mulcan2d  9418  mulcanad  9419  mulcan  9421  div11  9466  eqneg  9496  qredeq  12801  qredeu  12802  gexexlem  15160  prmirredlem  16462  tanarg  19986  quad2  20151  dcubic  20158  atandm2  20189  dvdsmulf1o  20450  dchrsum2  20523  sumdchr2  20525  lgseisenlem2  20605  2sqlem8  20627  ipasslem4  21428  ax5seg  24638  2wsms  25711  pell1234qrreccl  27042  pell14qrdich  27057  rmxyneg  27108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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