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Theorem mulcand 9397
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 ) )
Dummy variable  x is distinct from all other variables.

Proof of Theorem mulcand
StepHypRef Expression
1 mulcand.3 . . . 4  |-  ( ph  ->  C  e.  CC )
2 mulcand.4 . . . 4  |-  ( ph  ->  C  =/=  0 )
3 recex 9396 . . . 4  |-  ( ( C  e.  CC  /\  C  =/=  0 )  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
41, 2, 3syl2anc 644 . . 3  |-  ( ph  ->  E. x  e.  CC  ( C  x.  x
)  =  1 )
5 oveq2 5828 . . . . . 6  |-  ( ( C  x.  A )  =  ( C  x.  B )  ->  (
x  x.  ( C  x.  A ) )  =  ( x  x.  ( C  x.  B
) ) )
6 simprl 734 . . . . . . . . . . 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 8852 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  ( C  x.  x ) )
9 simprr 735 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( C  x.  x
)  =  1 )
108, 9eqtrd 2317 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  C
)  =  1 )
1110oveq1d 5835 . . . . . . . 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 8854 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  A
)  =  ( x  x.  ( C  x.  A ) ) )
1513mulid2d 8849 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  A
)  =  A )
1611, 14, 153eqtr3d 2325 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  A )
)  =  A )
1710oveq1d 5835 . . . . . . . 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 8854 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( ( x  x.  C )  x.  B
)  =  ( x  x.  ( C  x.  B ) ) )
2119mulid2d 8849 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( 1  x.  B
)  =  B )
2217, 20, 213eqtr3d 2325 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  CC  /\  ( C  x.  x )  =  1 ) )  -> 
( x  x.  ( C  x.  B )
)  =  B )
2316, 22eqeq12d 2299 . . . . . 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 600 . . . 4  |-  ( (
ph  /\  x  e.  CC )  ->  ( ( C  x.  x )  =  1  ->  (
( C  x.  A
)  =  ( C  x.  B )  ->  A  =  B )
) )
2625rexlimdva 2669 . . 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 5828 . 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 1624    e. wcel 1685    =/= wne 2448   E.wrex 2546  (class class class)co 5820   CCcc 8731   0cc0 8733   1c1 8734    x. cmul 8738
This theorem is referenced by:  mulcan2d  9398  mulcanad  9399  mulcan  9401  div11  9446  eqneg  9476  qredeq  12780  qredeu  12781  gexexlem  15139  prmirredlem  16441  tanarg  19965  quad2  20130  dcubic  20137  atandm2  20168  dvdsmulf1o  20429  dchrsum2  20502  sumdchr2  20504  lgseisenlem2  20584  2sqlem8  20606  ipasslem4  21405  ax5seg  23974  2wsms  25008  pell1234qrreccl  26339  pell14qrdich  26354  rmxyneg  26405
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-iota 6253  df-riota 6300  df-er 6656  df-en 6860  df-dom 6861  df-sdom 6862  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036
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