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Theorem divass 9652
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 9641 . . 3  |-  ( ( C  e.  CC  /\  C  =/=  0 )  -> 
( 1  /  C
)  e.  CC )
2 mulass 9034 . . 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 1219 . 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 9030 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
543adant3 977 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  x.  B
)  e.  CC )
6 simp3l 985 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  e.  CC )
7 simp3r 986 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  C  =/=  0 )
8 divrec 9650 . . 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 1184 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( A  x.  B )  x.  ( 1  /  C ) ) )
10 simp2 958 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
11 divrec 9650 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( B  /  C )  =  ( B  x.  (
1  /  C ) ) )
1210, 6, 7, 11syl3anc 1184 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1312oveq2d 6056 . 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 2446 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( A  x.  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    / cdiv 9633
This theorem is referenced by:  div23  9653  div32  9654  divasszi  9720  divassd  9781  lt2mul2div  9842  zdivmul  10298  mertenslem1  12616  efi4p  12693  divsqrsumlem  20771  basellem8  20823  logexprlim  20962  bposlem6  21026  lgsquadlem2  21092  chebbnd1lem3  21118  vmadivsum  21129  dchrmusum2  21141  dchrisum0lem1b  21162  dchrisum0lem2  21165  mudivsum  21177  mulog2sumlem2  21182  selberglem1  21192  selberglem2  21193  pntlemb  21244  pntlemr  21249  pntlemj  21250  pntlemf  21252  pntlemk  21253  pntlemo  21254  stoweidlem24  27640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634
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