Theorem add32 8134

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 8112	. . . 4 |- ( ( B e. CC /\ C e. CC ) -> ( B + C ) = ( C + B ) )
2	1	oveq2d 5907	. . 3 |- ( ( B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( A + ( C + B ) ) )
3	2	3adant1 1017	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A + ( B + C ) ) = ( A + ( C + B ) ) )
4		addass 7959	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( A + ( B + C ) ) )
5		addass 7959	. . 3 |- ( ( A e. CC /\ C e. CC /\ B e. CC ) -> ( ( A + C ) + B ) = ( A + ( C + B ) ) )
6	5	3com23 1211	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + C ) + B ) = ( A + ( C + B ) ) )
7	3, 4, 6	3eqtr4d 2232	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160  (class class class)co 5891   CCcc 7827   + caddc 7832

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcom 7929  ax-addass 7931

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5193  df-fv 5239  df-ov 5894

This theorem is referenced by:  add32r  8135  add32i  8139  add32d  8143  cnegexlem2  8151  cnegexlem3  8152  2addsub  8189  opeo  11920