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Theorem mul12 8932
Description: Commutative/associative law for multiplication. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
mul12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )

Proof of Theorem mul12
StepHypRef Expression
1 mulcom 8777 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
21oveq1d 5793 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  B )  x.  C
)  =  ( ( B  x.  A )  x.  C ) )
323adant3 980 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( B  x.  A )  x.  C ) )
4 mulass 8779 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulass 8779 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
653com12 1160 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  x.  A
)  x.  C )  =  ( B  x.  ( A  x.  C
) ) )
73, 4, 63eqtr3d 2296 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( B  x.  ( A  x.  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621  (class class class)co 5778   CCcc 8689    x. cmul 8696
This theorem is referenced by:  mul02  8944  mul12i  8961  mul12d  8975  mulre  11557  sqreulem  11794  demoivre  12428  demoivreALT  12429  dvdscmul  12503  dvdscmulr  12505  dvdstr  12510  ablfacrp  15249  nmoleub2lem3  18544  sinperlem  19796  coskpi  19836  sineq0  19837  efif1olem4  19855  rpvmasum2  20609  fsumcube  24156  expgrowthi  26903
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-mulcom 8755  ax-mulass 8757
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-rex 2522  df-rab 2525  df-v 2759  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fv 4675  df-ov 5781
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