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Theorem dmdcan 9557
Description: Cancellation law for division and multiplication. (Contributed by Scott Fenton, 7-Jun-2013.) (Proof shortened by Fan Zheng, 3-Jul-2016.)
Assertion
Ref Expression
dmdcan  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  (
( A  /  B
)  x.  ( C  /  A ) )  =  ( C  /  B ) )

Proof of Theorem dmdcan
StepHypRef Expression
1 simp1l 979 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  A  e.  CC )
2 simp3 957 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  C  e.  CC )
3 simp1r 980 . . . 4  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  A  =/=  0 )
4 divcl 9517 . . . 4  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( C  /  A )  e.  CC )
52, 1, 3, 4syl3anc 1182 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  ( C  /  A )  e.  CC )
6 simp2l 981 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  B  e.  CC )
7 simp2r 982 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  B  =/=  0 )
8 div23 9530 . . 3  |-  ( ( A  e.  CC  /\  ( C  /  A
)  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( ( A  x.  ( C  /  A ) )  /  B )  =  ( ( A  /  B
)  x.  ( C  /  A ) ) )
91, 5, 6, 7, 8syl112anc 1186 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  (
( A  x.  ( C  /  A ) )  /  B )  =  ( ( A  /  B )  x.  ( C  /  A ) ) )
10 divcan2 9519 . . . 4  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( A  x.  ( C  /  A ) )  =  C )
112, 1, 3, 10syl3anc 1182 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  ( A  x.  ( C  /  A ) )  =  C )
1211oveq1d 5957 . 2  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  (
( A  x.  ( C  /  A ) )  /  B )  =  ( C  /  B
) )
139, 12eqtr3d 2392 1  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  ( B  e.  CC  /\  B  =/=  0 )  /\  C  e.  CC )  ->  (
( A  /  B
)  x.  ( C  /  A ) )  =  ( C  /  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521  (class class class)co 5942   CCcc 8822   0cc0 8824    x. cmul 8829    / cdiv 9510
This theorem is referenced by:  dmdcand  9652  chtppilimlem2  20729  chebbnd2  20732  chpchtlim  20734  chpo1ub  20735  rplogsumlem2  20740  rpvmasumlem  20742  dchrisum0lem2a  20772  mulogsumlem  20786  pntibndlem2  20846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-po 4393  df-so 4394  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-riota 6388  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511
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