Theorem add4 7172

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 7169	. . . . 5 |- ( ( B e. CC /\ C e. CC /\ D e. CC ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
2	1	3expb 1105	. . . 4 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( B + ( C + D ) ) = ( C + ( B + D ) ) )
3	2	oveq2d 5528	. . 3 |- ( ( B e. CC /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
4	3	adantll 445	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A + ( B + ( C + D ) ) ) = ( A + ( C + ( B + D ) ) ) )
5		addcl 7006	. . 3 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) e. CC )
6		addass 7011	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
7	6	3expa 1104	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C + D ) e. CC ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
8	5, 7	sylan2 270	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) + ( C + D ) ) = ( A + ( B + ( C + D ) ) ) )
9		addcl 7006	. . . 4 |- ( ( B e. CC /\ D e. CC ) -> ( B + D ) e. CC )
10		addass 7011	. . . . 5 |- ( ( A e. CC /\ C e. CC /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
11	10	3expa 1104	. . . 4 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B + D ) e. CC ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
12	9, 11	sylan2 270	. . 3 |- ( ( ( A e. CC /\ C e. CC ) /\ ( B e. CC /\ D e. CC ) ) -> ( ( A + C ) + ( B + D ) ) = ( A + ( C + ( B + D ) ) ) )
13	12	an4s 522	. 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 2082	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 97   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   + caddc 6892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-addcl 6980  ax-addcom 6984  ax-addass 6986

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by:  add42  7173  add4i  7176  add4d  7180