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Theorem divass 10115
Description: An associative law for division. (Contributed by NM, 2-Aug-2004.)
Assertion
Ref Expression
divass  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( A  x.  ( B  /  C ) ) )

Proof of Theorem divass
StepHypRef Expression
1 reccl 10104 . . 3  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 1  /  C
)  e.  CC )
2 mulass 9473 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  (
1  /  C )  e.  CC )  -> 
( ( A  x.  B )  x.  (
1  /  C ) )  =  ( A  x.  ( B  x.  ( 1  /  C
) ) ) )
31, 2syl3an3 1254 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  x.  (
1  /  C ) )  =  ( A  x.  ( B  x.  ( 1  /  C
) ) ) )
4 mulcl 9469 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
543adant3 1008 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  x.  B
)  e.  CC )
6 simp3l 1016 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
7 simp3r 1017 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  =/=  0 )
8 divrec 10113 . . 3  |-  ( ( ( A  x.  B
)  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( A  x.  B
)  /  C )  =  ( ( A  x.  B )  x.  ( 1  /  C
) ) )
95, 6, 7, 8syl3anc 1219 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( A  x.  B )  x.  ( 1  /  C ) ) )
10 simp2 989 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
11 divrec 10113 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
1210, 6, 7, 11syl3anc 1219 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1312oveq2d 6208 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  x.  ( B  /  C ) )  =  ( A  x.  ( B  x.  (
1  /  C ) ) ) )
143, 9, 133eqtr4d 2502 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( A  x.  ( B  /  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6192   CCcc 9383   0cc0 9385   1c1 9386    x. cmul 9390    / cdiv 10096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097
This theorem is referenced by:  div23  10116  div32  10117  divasszi  10184  divassd  10245  lt2mul2div  10311  zdivmul  10817  mertenslem1  13448  efi4p  13525  divsqrsumlem  22491  basellem8  22543  logexprlim  22682  bposlem6  22746  lgsquadlem2  22812  chebbnd1lem3  22838  vmadivsum  22849  dchrmusum2  22861  dchrisum0lem1b  22882  dchrisum0lem2  22885  mudivsum  22897  mulog2sumlem2  22902  selberglem1  22912  selberglem2  22913  pntlemb  22964  pntlemr  22969  pntlemj  22970  pntlemf  22972  pntlemk  22973  pntlemo  22974  dvasin  28620  stoweidlem24  29959
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