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Theorem mul32i 8066
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 8049 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  x.  B
)  x.  C )  =  ( ( A  x.  C )  x.  B ) )
51, 2, 3, 4mp3an 1332 1  |-  ( ( A  x.  B )  x.  C )  =  ( ( A  x.  C )  x.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1348    e. wcel 2141  (class class class)co 5853   CCcc 7772    x. cmul 7779
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-mulcom 7875  ax-mulass 7877
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by:  8th4div3  9097
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