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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 9567 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 6285 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  /  C
)  =  ( ( B  x.  A )  /  C ) )
323adant3 1014 . 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 1198 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( ( B  x.  A )  /  C
)  =  ( B  x.  ( A  /  C ) ) )
6 simp2 995 . . 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 1195 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( A  /  C )  e.  CC )
983adant2 1013 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( A  /  C
)  e.  CC )
106, 9mulcomd 9606 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  -> 
( B  x.  ( A  /  C ) )  =  ( ( A  /  C )  x.  B ) )
113, 5, 103eqtrd 2499 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649  (class class class)co 6270   CCcc 9479   0cc0 9481    x. cmul 9486    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203
This theorem is referenced by:  div32  10223  div13  10224  divdiv32  10248  dmdcan  10250  div23i  10298  div23d  10353  digit2  12284  facdiv  12350  mertenslem1  13778  bposlem9  23768  lgsquadlem2  23831  chtppilimlem2  23860  vmadivsum  23868  dchrmusum2  23880  dchrvmasumlem1  23881  dchrvmasumlem2  23884  mudivsum  23916  mulog2sumlem2  23921  selberglem1  23931  selberglem2  23932  selberg2lem  23936  pntibndlem2  23977  pntlemb  23983  pntlemn  23986  pntlemr  23988  pntlemj  23989  pntlemf  23991  pntlemk  23992  pntlemo  23993  siii  25969  riesz3i  27182  subdivcomb2  29350  dignn0fr  33495
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