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Theorem subdi 8170
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 982 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp3 984 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
3 subcl 7984 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
433adant1 1000 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
51, 2, 4adddid 7813 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( ( A  x.  C )  +  ( A  x.  ( B  -  C )
) ) )
6 pncan3 7993 . . . . . . 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 1000 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
98oveq2d 5797 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( A  x.  B ) )
105, 9eqtr3d 2175 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
)  +  ( A  x.  ( B  -  C ) ) )  =  ( A  x.  B ) )
11 mulcl 7770 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
12113adant3 1002 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
13 mulcl 7770 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
14133adant2 1001 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
15 mulcl 7770 . . . . . 6  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
163, 15sylan2 284 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
17163impb 1178 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  e.  CC )
1812, 14, 17subaddd 8114 . . 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 2146 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 963    = wceq 1332    e. wcel 1481  (class class class)co 5781   CCcc 7641    + caddc 7646    x. cmul 7648    - cmin 7956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-setind 4459  ax-resscn 7735  ax-1cn 7736  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-addcom 7743  ax-addass 7745  ax-distr 7747  ax-i2m1 7748  ax-0id 7751  ax-rnegex 7752  ax-cnre 7754
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-sub 7958
This theorem is referenced by:  subdir  8171  subdii  8192  subdid  8199  expubnd  10380  subsq  10429  cos01bnd  11499  modmulconst  11559  odd2np1  11604  omoe  11627  omeo  11629  phiprmpw  11932
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