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Theorem subdivcomb2 27314
Description: Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
Assertion
Ref Expression
subdivcomb2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  -  ( C  x.  B
) )  /  C
)  =  ( ( A  /  C )  -  B ) )

Proof of Theorem subdivcomb2
StepHypRef Expression
1 simp3l 1011 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
2 simp2 984 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
31, 2mulcld 9402 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( C  x.  B
)  e.  CC )
4 divsubdir 10023 . . 3  |-  ( ( A  e.  CC  /\  ( C  x.  B
)  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( A  -  ( C  x.  B ) )  /  C )  =  ( ( A  /  C
)  -  ( ( C  x.  B )  /  C ) ) )
53, 4syld3an2 1260 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  -  ( C  x.  B
) )  /  C
)  =  ( ( A  /  C )  -  ( ( C  x.  B )  /  C ) ) )
6 simprl 750 . . . . . 6  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
7 simpl 454 . . . . . 6  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
8 simpr 458 . . . . . 6  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( C  e.  CC  /\  C  =/=  0 ) )
9 div23 10009 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  B )  /  C
)  =  ( ( C  /  C )  x.  B ) )
106, 7, 8, 9syl3anc 1213 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  B )  /  C )  =  ( ( C  /  C
)  x.  B ) )
11 divid 10017 . . . . . . 7  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( C  /  C
)  =  1 )
1211oveq1d 6105 . . . . . 6  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( ( C  /  C )  x.  B
)  =  ( 1  x.  B ) )
13 mulid2 9380 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
1412, 13sylan9eqr 2495 . . . . 5  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  /  C )  x.  B )  =  B )
1510, 14eqtrd 2473 . . . 4  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  B )  /  C )  =  B )
16153adant1 1001 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
1716oveq2d 6106 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  /  C )  -  (
( C  x.  B
)  /  C ) )  =  ( ( A  /  C )  -  B ) )
185, 17eqtrd 2473 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  -  ( C  x.  B
) )  /  C
)  =  ( ( A  /  C )  -  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    x. cmul 9283    - cmin 9591    / cdiv 9989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990
This theorem is referenced by: (None)
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