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Theorem subdi 9467
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 957 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp3 959 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
3 subcl 9305 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
433adant1 975 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
51, 2, 4adddid 9112 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( ( A  x.  C )  +  ( A  x.  ( B  -  C )
) ) )
6 pncan3 9313 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
76ancoms 440 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
873adant1 975 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
98oveq2d 6097 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( A  x.  B ) )
105, 9eqtr3d 2470 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
)  +  ( A  x.  ( B  -  C ) ) )  =  ( A  x.  B ) )
11 mulcl 9074 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
12113adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
13 mulcl 9074 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
14133adant2 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
15 mulcl 9074 . . . . . 6  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
163, 15sylan2 461 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
17163impb 1149 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  e.  CC )
1812, 14, 17subaddd 9429 . . 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 224 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  -  ( A  x.  C ) )  =  ( A  x.  ( B  -  C
) ) )
2019eqcomd 2441 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988    + caddc 8993    x. cmul 8995    - cmin 9291
This theorem is referenced by:  subdir  9468  subdii  9482  subdid  9489  expubnd  11440  subsq  11488  cos01bnd  12787  odd2np1  12908  phiprmpw  13165  omoe  13186  omeo  13188  pythagtriplem14  13202  plydiveu  20215  quotcan  20226  basellem9  20871  chtublem  20995  bposlem9  21076  ipval2  22203  mulcan1g  25189  ax5seglem1  25867  ax5seglem2  25868  axpaschlem  25879  axcontlem2  25904  axcontlem4  25906  axcontlem7  25909  axcontlem8  25910  bpoly3  26104  ftc1anclem6  26285  pellexlem6  26897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-ltxr 9125  df-sub 9293
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