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Theorem add32r 7405
Description: Commutative/associative law that swaps the last two terms in a triple sum, rearranging the parentheses. (Contributed by Paul Chapman, 18-May-2007.)
Assertion
Ref Expression
add32r  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C ) )  =  ( ( A  +  C )  +  B
) )

Proof of Theorem add32r
StepHypRef Expression
1 addass 7235 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
2 add32 7404 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( ( A  +  C )  +  B ) )
31, 2eqtr3d 2117 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C ) )  =  ( ( A  +  C )  +  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 920    = wceq 1285    e. wcel 1434  (class class class)co 5564   CCcc 7111    + caddc 7116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-addcom 7208  ax-addass 7210
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5567
This theorem is referenced by: (None)
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