Theorem add4 7923

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 7920	. . . . 5 |- ( ( B e. CC /\ C e. CC /\ D e. CC ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
2	1	3expb 1182	. . . 4 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
3	2	oveq2d 5790	. . 3 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
4	3	adantll 467	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
5		addcl 7745	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
6		addass 7750	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
7	6	3expa 1181	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
8	5, 7	sylan2 284	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
9		addcl 7745	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) e. CC )
10		addass 7750	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
11	10	3expa 1181	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
12	9, 11	sylan2 284	. . 3 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
13	12	an4s 577	. 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 2182	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 103   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   + caddc 7623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-addcl 7716  ax-addcom 7720  ax-addass 7722

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777

This theorem is referenced by:  add42  7924  add4i  7927  add4d  7931  3dvds2dec  11563  opoe  11592  ptolemy  12905