Theorem addsub4 8389

Description: Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by NM, 4-Mar-2005.)

Assertion

Ref	Expression
addsub4	|- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) - ( C + D ) ) = ( ( A - C ) + ( B - D ) ) )

Proof of Theorem addsub4

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> A e. CC )
2		simplr 528	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> B e. CC )
3		simprl 529	. . . 4 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> C e. CC )
4		addsub 8357	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) - C ) = ( ( A - C ) + B ) )
5	1, 2, 3, 4	syl3anc 1271	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( A + B ) - C ) = ( ( A - C ) + B ) )
6	5	oveq1d 6016	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + B ) - C ) - D ) = ( ( ( A - C ) + B ) - D ) )
7	1, 2	addcld 8166	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A + B ) e. CC )
8		simprr 531	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> D e. CC )
9		subsub4 8379	. . 3 |- ( ( ( A + B ) e. CC /\ C e. CC /\ D e. CC ) -> ( ( ( A + B ) - C ) - D ) = ( ( A + B ) - ( C + D ) ) )
10	7, 3, 8, 9	syl3anc 1271	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A + B ) - C ) - D ) = ( ( A + B ) - ( C + D ) ) )
11		subcl 8345	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( A - C ) e. CC )
12	11	ad2ant2r 509	. . 3 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( A - C ) e. CC )
13		addsubass 8356	. . 3 |- ( ( ( A - C ) e. CC /\ B e. CC /\ D e. CC ) -> ( ( ( A - C ) + B ) - D ) = ( ( A - C ) + ( B - D ) ) )
14	12, 2, 8, 13	syl3anc 1271	. 2 |- ( ( ( A e. CC /\ B e. CC ) /\ ( C e. CC /\ D e. CC ) ) -> ( ( ( A - C ) + B ) - D ) = ( ( A - C ) + ( B - D ) ) )
15	6, 10, 14	3eqtr3d 2270	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 1395   e. wcel 2200  (class class class)co 6001   CCcc 7997   + caddc 8002   - cmin 8317

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629  ax-resscn 8091  ax-1cn 8092  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-sub 8319

This theorem is referenced by:  subadd4  8390  addsub4i  8442  addsub4d  8504  ser3sub  10745