Theorem add4 8450

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 8447	. . . . 5 |- ( ( B e. CC /\ C e. CC /\ D e. CC ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
2	1	3expb 1231	. . . 4 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
3	2	oveq2d 6074	. . 3 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
4	3	adantll 476	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
5		addcl 8268	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
6		addass 8273	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
7	6	3expa 1230	. . 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 8268	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) e. CC )
10		addass 8273	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
11	10	3expa 1230	. . . 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 592	. 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 2277	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 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-addcl 8239  ax-addcom 8243  ax-addass 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  add42  8451  add4i  8454  add4d  8458  3dvds2dec  12577  opoe  12606  ptolemy  15815