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Theorem subdi 9181
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 960 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
2 simp3 962 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
3 subcl 9019 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C
)  e.  CC )
433adant1 978 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  -  C )  e.  CC )
51, 2, 4adddid 8827 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( ( A  x.  C )  +  ( A  x.  ( B  -  C )
) ) )
6 pncan3 9027 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
76ancoms 441 . . . . . 6  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
873adant1 978 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  +  ( B  -  C ) )  =  B )
98oveq2d 5808 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( C  +  ( B  -  C ) ) )  =  ( A  x.  B ) )
105, 9eqtr3d 2292 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  C
)  +  ( A  x.  ( B  -  C ) ) )  =  ( A  x.  B ) )
11 mulcl 8789 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
12113adant3 980 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
13 mulcl 8789 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
14133adant2 979 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
15 mulcl 8789 . . . . . 6  |-  ( ( A  e.  CC  /\  ( B  -  C
)  e.  CC )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
163, 15sylan2 462 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  C  e.  CC ) )  ->  ( A  x.  ( B  -  C
) )  e.  CC )
17163impb 1152 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  -  C ) )  e.  CC )
1812, 14, 17subaddd 9143 . . 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 225 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  -  ( A  x.  C ) )  =  ( A  x.  ( B  -  C
) ) )
2019eqcomd 2263 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5792   CCcc 8703    + caddc 8708    x. cmul 8710    - cmin 9005
This theorem is referenced by:  subdir  9182  subdii  9196  subdid  9203  expubnd  11129  subsq  11177  cos01bnd  12429  odd2np1  12550  phiprmpw  12807  omoe  12828  omeo  12830  pythagtriplem14  12844  plydiveu  19641  quotcan  19652  basellem9  20289  chtublem  20413  bposlem9  20494  ipval2  21241  mulcan1g  23456  ax5seglem1  23932  ax5seglem2  23933  axpaschlem  23944  axcontlem2  23969  axcontlem4  23971  axcontlem7  23974  axcontlem8  23975  bpoly3  24169  fsumcube  24171  pellexlem6  26287  stoweidlem26  27144
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-ltxr 8840  df-sub 9007
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