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Theorem mul32i 7877
Description: Commutative/associative law that swaps the last two factors in a triple product. (Contributed by NM, 11-May-1999.)
Hypotheses
Ref Expression
mul.1  |-  A  e.  CC
mul.2  |-  B  e.  CC
mul.3  |-  C  e.  CC
Assertion
Ref Expression
mul32i  |-  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B
)

Proof of Theorem mul32i
StepHypRef Expression
1 mul.1 . 2  |-  A  e.  CC
2 mul.2 . 2  |-  B  e.  CC
3 mul.3 . 2  |-  C  e.  CC
4 mul32 7860 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4mp3an 1300 1  |-  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465  (class class class)co 5742   CCcc 7586    x. cmul 7593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-mulcom 7689  ax-mulass 7691
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  8th4div3  8907
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