Theorem addcnsr 9956

Description: Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
addcnsr	|- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. )

Proof of Theorem addcnsr

Dummy variables x y z w v u f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 4932	. 2 |- <. ( A +R C ) , ( B +R D ) >. e. _V
2		oveq1 6657	. . . 4 |- ( w = A -> ( w +R u ) = ( A +R u ) )
3		oveq1 6657	. . . 4 |- ( v = B -> ( v +R f ) = ( B +R f ) )
4		opeq12 4404	. . . 4 |- ( ( ( w +R u ) = ( A +R u ) /\ ( v +R f ) = ( B +R f ) ) -> <. ( w +R u ) , ( v +R f ) >. = <. ( A +R u ) , ( B +R f ) >. )
5	2, 3, 4	syl2an 494	. . 3 |- ( ( w = A /\ v = B ) -> <. ( w +R u ) , ( v +R f ) >. = <. ( A +R u ) , ( B +R f ) >. )
6		oveq2 6658	. . . 4 |- ( u = C -> ( A +R u ) = ( A +R C ) )
7		oveq2 6658	. . . 4 |- ( f = D -> ( B +R f ) = ( B +R D ) )
8		opeq12 4404	. . . 4 |- ( ( ( A +R u ) = ( A +R C ) /\ ( B +R f ) = ( B +R D ) ) -> <. ( A +R u ) , ( B +R f ) >. = <. ( A +R C ) , ( B +R D ) >. )
9	6, 7, 8	syl2an 494	. . 3 |- ( ( u = C /\ f = D ) -> <. ( A +R u ) , ( B +R f ) >. = <. ( A +R C ) , ( B +R D ) >. )
10	5, 9	sylan9eq 2676	. 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( w +R u ) , ( v +R f ) >. = <. ( A +R C ) , ( B +R D ) >. )
11		df-add 9947	. . 3 |- + = { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
12		df-c 9942	. . . . . . 7 |- CC = ( R. X. R. )
13	12	eleq2i 2693	. . . . . 6 |- ( x e. CC <-> x e. ( R. X. R. ) )
14	12	eleq2i 2693	. . . . . 6 |- ( y e. CC <-> y e. ( R. X. R. ) )
15	13, 14	anbi12i 733	. . . . 5 |- ( ( x e. CC /\ y e. CC ) <-> ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) )
16	15	anbi1i 731	. . . 4 |- ( ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) <-> ( ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) )
17	16	oprabbii 6710	. . 3 |- { <. <. x , y >. , z >. | ( ( x e. CC /\ y e. CC ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) } = { <. <. x , y >. , z >. | ( ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
18	11, 17	eqtri 2644	. 2 |- + = { <. <. x , y >. , z >. | ( ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = <. ( w +R u ) , ( v +R f ) >. ) ) }
19	1, 10, 18	ov3 6797	1 |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   X. cxp 5112  (class class class)co 6650   {coprab 6651   R.cnr 9687   +R cplr 9691   CCcc 9934   + caddc 9939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-c 9942  df-add 9947

This theorem is referenced by:  addresr  9959  addcnsrec  9964  axaddf  9966  axcnre  9985