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Theorem div23 10222
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 9574 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6297 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  /  C
)  =  ( ( B  x.  A )  /  C ) )
323adant3 1016 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( B  x.  A )  /  C ) )
4 divass 10221 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( B  x.  A )  /  C
)  =  ( B  x.  ( A  /  C ) ) )
543com12 1200 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( B  x.  A )  /  C
)  =  ( B  x.  ( A  /  C ) ) )
6 simp2 997 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  B  e.  CC )
7 divcl 10209 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  ( A  /  C )  e.  CC )
873expb 1197 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
983adant2 1015 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
106, 9mulcomd 9613 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  x.  ( A  /  C ) )  =  ( ( A  /  C )  x.  B ) )
113, 5, 103eqtrd 2512 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( A  x.  B )  /  C
)  =  ( ( A  /  C )  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6282   CCcc 9486   0cc0 9488    x. cmul 9493    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203
This theorem is referenced by:  div32  10223  div13  10224  divdiv32  10248  dmdcan  10250  div23i  10298  div23d  10353  digit2  12263  facdiv  12329  mertenslem1  13652  bposlem9  23295  lgsquadlem2  23358  chtppilimlem2  23387  vmadivsum  23395  dchrmusum2  23407  dchrvmasumlem1  23408  dchrvmasumlem2  23411  mudivsum  23443  mulog2sumlem2  23448  selberglem1  23458  selberglem2  23459  selberg2lem  23463  pntibndlem2  23504  pntlemb  23510  pntlemn  23513  pntlemr  23515  pntlemj  23516  pntlemf  23518  pntlemk  23519  pntlemo  23520  siii  25444  riesz3i  26657  subdivcomb2  28581
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