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Theorem mulcand 9369
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
StepHypRef Expression
1 mulcand.3 . . . 4  |-  ( ph  ->  C  e.  CC )
2 mulcand.4 . . . 4  |-  ( ph  ->  C  =/=  0 )
3 recex 9368 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
41, 2, 3syl2anc 645 . . 3  |-  ( ph  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
5 oveq2 5800 . . . . . 6  |-  ( ( C  x.  A )  =  ( C  x.  B )  ->  (
x  x.  ( C  x.  A ) )  =  ( x  x.  ( C  x.  B
) ) )
6 simprl 735 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  x  e.  CC )
71adantr 453 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  C  e.  CC )
86, 7mulcomd 8824 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  ( C  x.  x ) )
9 simprr 736 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( C  x.  x
)  =  1 )
108, 9eqtrd 2290 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  1 )
1110oveq1d 5807 . . . . . . . 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 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  A  e.  CC )
146, 7, 13mulassd 8826 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( x  x.  ( C  x.  A ) ) )
1513mulid2d 8821 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
1611, 14, 153eqtr3d 2298 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  A )
)  =  A )
1710oveq1d 5807 . . . . . . . 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 453 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  ->  B  e.  CC )
206, 7, 19mulassd 8826 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( x  x.  ( C  x.  B ) ) )
2119mulid2d 8821 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  B
)  =  B )
2217, 20, 213eqtr3d 2298 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  B )
)  =  B )
2316, 22eqeq12d 2272 . . . . . 6  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  ( C  x.  A
) )  =  ( x  x.  ( C  x.  B ) )  <-> 
A  =  B ) )
245, 23syl5ib 212 . . . . 5  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
2524expr 601 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( C  x.  x )  =  1  ->  (
( C  x.  A
)  =  ( C  x.  B )  ->  A  =  B )
) )
2625rexlimdva 2642 . . 3  |-  ( ph  ->  ( E. x  e.  CC  ( C  x.  x )  =  1  ->  ( ( C  x.  A )  =  ( C  x.  B
)  ->  A  =  B ) ) )
274, 26mpd 16 . 2  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  ->  A  =  B ) )
28 oveq2 5800 . 2  |-  ( A  =  B  ->  ( C  x.  A )  =  ( C  x.  B ) )
2927, 28impbid1 196 1  |-  ( ph  ->  ( ( C  x.  A )  =  ( C  x.  B )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519  (class class class)co 5792   CCcc 8703   0cc0 8705   1c1 8706    x. cmul 8710
This theorem is referenced by:  mulcan2d  9370  mulcanad  9371  mulcan  9373  div11  9418  eqneg  9448  qredeq  12747  qredeu  12748  gexexlem  15106  prmirredlem  16408  tanarg  19932  quad2  20097  dcubic  20104  atandm2  20135  dvdsmulf1o  20396  dchrsum2  20469  sumdchr2  20471  lgseisenlem2  20551  2sqlem8  20573  ipasslem4  21372  ax5seg  23941  2wsms  24975  pell1234qrreccl  26306  pell14qrdich  26321  rmxyneg  26372
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008
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