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Theorem add12 7638
Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by NM, 11-May-2004.)
Assertion
Ref Expression
add12  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C ) ) )

Proof of Theorem add12
StepHypRef Expression
1 addcom 7617 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
21oveq1d 5667 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  +  C
)  =  ( ( B  +  A )  +  C ) )
323adant3 963 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( ( B  +  A )  +  C ) )
4 addass 7470 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  +  C )  =  ( A  +  ( B  +  C
) ) )
5 addass 7470 . . 3  |-  ( ( B  e.  CC  /\  A  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  +  C )  =  ( B  +  ( A  +  C
) ) )
653com12 1147 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( B  +  A
)  +  C )  =  ( B  +  ( A  +  C
) ) )
73, 4, 63eqtr3d 2128 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  +  ( B  +  C ) )  =  ( B  +  ( A  +  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7346    + caddc 7351
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-addcom 7443  ax-addass 7445
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-iota 4980  df-fv 5023  df-ov 5655
This theorem is referenced by:  add4  7641  add12i  7643  add12d  7647
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