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Theorem subdir 8172
Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18. (Contributed by NM, 30-Dec-2005.)
Assertion
Ref Expression
subdir  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  x.  C )  =  ( ( A  x.  C )  -  ( B  x.  C
) ) )

Proof of Theorem subdir
StepHypRef Expression
1 subdi 8171 . . 3  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  B  e.  CC )  ->  ( C  x.  ( A  -  B ) )  =  ( ( C  x.  A )  -  ( C  x.  B )
) )
213coml 1189 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  ( A  -  B ) )  =  ( ( C  x.  A )  -  ( C  x.  B )
) )
3 subcl 7985 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
4 mulcom 7773 . . . 4  |-  ( ( ( A  -  B
)  e.  CC  /\  C  e.  CC )  ->  ( ( A  -  B )  x.  C
)  =  ( C  x.  ( A  -  B ) ) )
53, 4sylan 281 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  C  e.  CC )  ->  ( ( A  -  B )  x.  C )  =  ( C  x.  ( A  -  B ) ) )
653impa 1177 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  x.  C )  =  ( C  x.  ( A  -  B
) ) )
7 mulcom 7773 . . . 4  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
873adant2 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
9 mulcom 7773 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 1000 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
118, 10oveq12d 5800 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
)  -  ( B  x.  C ) )  =  ( ( C  x.  A )  -  ( C  x.  B
) ) )
122, 6, 113eqtr4d 2183 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  -  B
)  x.  C )  =  ( ( A  x.  C )  -  ( B  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 5782   CCcc 7642    x. cmul 7649    - cmin 7957
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 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460  ax-resscn 7736  ax-1cn 7737  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755
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 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-sub 7959
This theorem is referenced by:  mul02  8173  mulneg1  8181  subdiri  8194  subdird  8201  dvds2sub  11564  cncongr1  11820  cncongr2  11821
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