Theorem add4 8477

Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
add4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )

Proof of Theorem add4

Step	Hyp	Ref	Expression
1		add12 8474	. . . . 5 |- ( ( B e. CC /\ C e. CC /\ D e. CC ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
2	1	3expb 1235	. . . 4 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
3	2	oveq2d 6091	. . 3 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
4	3	adantll 480	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
5		addcl 8294	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
6		addass 8299	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
7	6	3expa 1234	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
8	5, 7	sylan2 286	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
9		addcl 8294	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) e. CC )
10		addass 8299	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
11	10	3expa 1234	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
12	9, 11	sylan2 286	. . 3 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
13	12	an4s 596	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
14	4, 8, 13	3eqtr4d 2281	1 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( ( A + C ) + ( B + D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   + caddc 8172

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-addcl 8265  ax-addcom 8269  ax-addass 8271

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  add42  8478  add4i  8481  add4d  8485  3dvds2dec  12611  opoe  12640  ptolemy  15848