Theorem add32i 8266

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by NM, 21-Jan-1997.)

Hypotheses

Ref	Expression
add.1	|- A e. CC
add.2	|- B e. CC
add.3	|- C e. CC

Assertion

Ref	Expression
add32i	|- ( ( A + B ) + C ) = ( ( A + C ) + B )

Proof of Theorem add32i

Step	Hyp	Ref	Expression
1		add.1	. 2 |- A e. CC
2		add.2	. 2 |- B e. CC
3		add.3	. 2 |- C e. CC
4		add32 8261	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) + C ) = ( ( A + C ) + B ) )
5	1, 2, 3, 4	mp3an 1350	1 |- ( ( A + B ) + C ) = ( ( A + C ) + B )

Syntax hints:   = wceq 1373   e. wcel 2177  (class class class)co 5962   CCcc 7953   + caddc 7958

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-addcom 8055  ax-addass 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-iota 5246  df-fv 5293  df-ov 5965

This theorem is referenced by:  karatsuba  12838  lgsdir2lem2  15591