Theorem add32 8116

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 13-Nov-1999.)

Assertion

Ref	Expression
add32	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Proof of Theorem add32

Step	Hyp	Ref	Expression
1		addcom 8094	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
2	1	oveq2d 5891	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( A + ( C + B ) ) )
3	2	3adant1 1015	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( A + ( C + B ) ) )
4		addass 7941	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5		addass 7941	. . 3 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) + B ) = ( A + ( C + B ) ) )
6	5	3com23 1209	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) + B ) = ( A + ( C + B ) ) )
7	3, 4, 6	3eqtr4d 2220	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148  (class class class)co 5875   CCcc 7809   + caddc 7814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-addcom 7911  ax-addass 7913

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878

This theorem is referenced by:  add32r  8117  add32i  8121  add32d  8125  cnegexlem2  8133  cnegexlem3  8134  2addsub  8171  opeo  11902