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Theorem div23 9689
Description: A commutative/associative law for division. (Contributed by NM, 2-Aug-2004.)
Assertion
Ref Expression
div23  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( A  /  C )  x.  B ) )

Proof of Theorem div23
StepHypRef Expression
1 mulcom 9068 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6088 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  /  C
)  =  ( ( B  x.  A )  /  C ) )
323adant3 977 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( B  x.  A )  /  C ) )
4 divass 9688 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( B  x.  A )  /  C
)  =  ( B  x.  ( A  /  C ) ) )
543com12 1157 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( B  x.  A )  /  C
)  =  ( B  x.  ( A  /  C ) ) )
6 simp2 958 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
7 divcl 9676 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
873expb 1154 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
983adant2 976 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
106, 9mulcomd 9101 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  x.  ( A  /  C ) )  =  ( ( A  /  C )  x.  B ) )
113, 5, 103eqtrd 2471 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( A  /  C )  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598  (class class class)co 6073   CCcc 8980   0cc0 8982    x. cmul 8987    / cdiv 9669
This theorem is referenced by:  div32  9690  div13  9691  divdiv32  9714  dmdcan  9716  div23i  9764  div23d  9819  digit2  11504  facdiv  11570  mertenslem1  12653  bposlem9  21068  lgsquadlem2  21131  chtppilimlem2  21160  vmadivsum  21168  dchrmusum2  21180  dchrvmasumlem1  21181  dchrvmasumlem2  21184  mudivsum  21216  mulog2sumlem2  21221  selberglem1  21231  selberglem2  21232  selberg2lem  21236  pntibndlem2  21277  pntlemb  21283  pntlemn  21286  pntlemr  21288  pntlemj  21289  pntlemf  21291  pntlemk  21292  pntlemo  21293  siii  22346  riesz3i  23557  subdivcomb2  25188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670
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