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Theorem apmul1 8775
Description: Multiplication of both sides of complex apartness by a complex number apart from zero. (Contributed by Jim Kingdon, 20-Mar-2020.)
Assertion
Ref Expression
apmul1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )

Proof of Theorem apmul1
StepHypRef Expression
1 simp1 999 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  A  e.  CC )
2 simp3l 1027 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C  e.  CC )
3 simp3r 1028 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  C #  0 )
42, 3recclapd 8768 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( 1  /  C
)  e.  CC )
51, 2, 4mulassd 8011 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  C )  x.  (
1  /  C ) )  =  ( A  x.  ( C  x.  ( 1  /  C
) ) ) )
62, 3recidapd 8770 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( C  x.  (
1  /  C ) )  =  1 )
76oveq2d 5912 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  ( C  x.  ( 1  /  C ) ) )  =  ( A  x.  1 ) )
81mulridd 8004 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  1 )  =  A )
95, 7, 83eqtrd 2226 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  C )  x.  (
1  /  C ) )  =  A )
10 simp2 1000 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  B  e.  CC )
1110, 2, 4mulassd 8011 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( B  x.  C )  x.  (
1  /  C ) )  =  ( B  x.  ( C  x.  ( 1  /  C
) ) ) )
126oveq2d 5912 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( B  x.  ( C  x.  ( 1  /  C ) ) )  =  ( B  x.  1 ) )
1310mulridd 8004 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( B  x.  1 )  =  B )
1411, 12, 133eqtrd 2226 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( B  x.  C )  x.  (
1  /  C ) )  =  B )
159, 14breq12d 4031 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( A  x.  C )  x.  ( 1  /  C
) ) #  ( ( B  x.  C )  x.  ( 1  /  C ) )  <->  A #  B
) )
161, 2mulcld 8008 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A  x.  C
)  e.  CC )
1710, 2mulcld 8008 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( B  x.  C
)  e.  CC )
18 mulext1 8599 . . . 4  |-  ( ( ( A  x.  C
)  e.  CC  /\  ( B  x.  C
)  e.  CC  /\  ( 1  /  C
)  e.  CC )  ->  ( ( ( A  x.  C )  x.  ( 1  /  C ) ) #  ( ( B  x.  C
)  x.  ( 1  /  C ) )  ->  ( A  x.  C ) #  ( B  x.  C ) ) )
1916, 17, 4, 18syl3anc 1249 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( ( A  x.  C )  x.  ( 1  /  C
) ) #  ( ( B  x.  C )  x.  ( 1  /  C ) )  -> 
( A  x.  C
) #  ( B  x.  C ) ) )
2015, 19sylbird 170 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A #  B  -> 
( A  x.  C
) #  ( B  x.  C ) ) )
21 mulext1 8599 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
) #  ( B  x.  C )  ->  A #  B ) )
22213adant3r 1237 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( ( A  x.  C ) #  ( B  x.  C )  ->  A #  B ) )
2320, 22impbid 129 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  -> 
( A #  B  <->  ( A  x.  C ) #  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    e. wcel 2160   class class class wbr 4018  (class class class)co 5896   CCcc 7839   0cc0 7841   1c1 7842    x. cmul 7846   # cap 8568    / cdiv 8659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-mulrcl 7940  ax-addcom 7941  ax-mulcom 7942  ax-addass 7943  ax-mulass 7944  ax-distr 7945  ax-i2m1 7946  ax-0lt1 7947  ax-1rid 7948  ax-0id 7949  ax-rnegex 7950  ax-precex 7951  ax-cnre 7952  ax-pre-ltirr 7953  ax-pre-ltwlin 7954  ax-pre-lttrn 7955  ax-pre-apti 7956  ax-pre-ltadd 7957  ax-pre-mulgt0 7958  ax-pre-mulext 7959
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-xr 8026  df-ltxr 8027  df-le 8028  df-sub 8160  df-neg 8161  df-reap 8562  df-ap 8569  df-div 8660
This theorem is referenced by:  apmul2  8776  divap1d  8788  apdivmuld  8800  qapne  9669  apcxp2  14815
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