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Theorem subdi 8060
Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 18-Nov-2004.)
Assertion
Ref Expression
subdi  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C )
) )

Proof of Theorem subdi
StepHypRef Expression
1 simp1 962 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp3 964 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
3 subcl 7878 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
433adant1 980 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
51, 2, 4adddid 7708 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( ( A  x.  C )  +  ( A  x.  ( B  -  C )
) ) )
6 pncan3 7887 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
76ancoms 266 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
873adant1 980 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
98oveq2d 5742 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( A  x.  B ) )
105, 9eqtr3d 2147 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
)  +  ( A  x.  ( B  -  C ) ) )  =  ( A  x.  B ) )
11 mulcl 7665 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
12113adant3 982 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
13 mulcl 7665 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
14133adant2 981 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
15 mulcl 7665 . . . . . 6  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
163, 15sylan2 282 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
17163impb 1158 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  e.  CC )
1812, 14, 17subaddd 8008 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( ( A  x.  B )  -  ( A  x.  C )
)  =  ( A  x.  ( B  -  C ) )  <->  ( ( A  x.  C )  +  ( A  x.  ( B  -  C
) ) )  =  ( A  x.  B
) ) )
1910, 18mpbird 166 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  -  ( A  x.  C ) )  =  ( A  x.  ( B  -  C
) ) )
2019eqcomd 2118 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  =  ( ( A  x.  B )  -  ( A  x.  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 943    = wceq 1312    e. wcel 1461  (class class class)co 5726   CCcc 7539    + caddc 7544    x. cmul 7546    - cmin 7850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410  ax-resscn 7631  ax-1cn 7632  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-sub 7852
This theorem is referenced by:  subdir  8061  subdii  8082  subdid  8089  expubnd  10237  subsq  10286  cos01bnd  11310  modmulconst  11367  odd2np1  11412  omoe  11435  omeo  11437  phiprmpw  11737
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