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Theorem mul31 9737
Description: Commutative/associative law. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
mul31  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( C  x.  B )  x.  A ) )

Proof of Theorem mul31
StepHypRef Expression
1 mulcom 9567 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
21oveq2d 6286 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C )
)  =  ( A  x.  ( C  x.  B ) ) )
323adant1 1012 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  x.  C ) )  =  ( A  x.  ( C  x.  B )
) )
4 mulass 9569 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( A  x.  ( B  x.  C
) ) )
5 mulcl 9565 . . . . 5  |-  ( ( C  e.  CC  /\  B  e.  CC )  ->  ( C  x.  B
)  e.  CC )
65ancoms 451 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B
)  e.  CC )
763adant1 1012 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( C  x.  B )  e.  CC )
8 simp1 994 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  A  e.  CC )
97, 8mulcomd 9606 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( C  x.  B
)  x.  A )  =  ( A  x.  ( C  x.  B
) ) )
103, 4, 93eqtr4d 2505 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( C  x.  B )  x.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823  (class class class)co 6270   CCcc 9479    x. cmul 9486
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-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-mulcl 9543  ax-mulcom 9545  ax-mulass 9547
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  mul02lem1  9745  addid1  9749  mul31d  9780
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