Theorem add4 7336

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 7333	. . . . 5 |- ( ( B e. CC /\ C e. CC /\ D e. CC ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
2	1	3expb 1140	. . . 4 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
3	2	oveq2d 5559	. . 3 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
4	3	adantll 460	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
5		addcl 7160	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
6		addass 7165	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
7	6	3expa 1139	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
8	5, 7	sylan2 280	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
9		addcl 7160	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) e. CC )
10		addass 7165	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
11	10	3expa 1139	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
12	9, 11	sylan2 280	. . 3 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
13	12	an4s 553	. 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 2124	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 102   = wceq 1285   e. wcel 1434  (class class class)co 5543   CCcc 7041   + caddc 7046

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-addcl 7134  ax-addcom 7138  ax-addass 7140

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546

This theorem is referenced by:  add42  7337  add4i  7340  add4d  7344  3dvds2dec  10410  opoe  10439