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Metamath Proof Explorer

Theorem addcnsr

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

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

Proof

Step Hyp Ref Expression
1 opex 
 |-  <. ( A +R C ) , ( B +R D ) >. e. _V
2 oveq1 
 |-  ( w = A -> ( w +R u ) = ( A +R u ) )
3 oveq1 
 |-  ( v = B -> ( v +R f ) = ( B +R f ) )
4 opeq12 
 |-  ( ( ( 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 
 |-  ( ( w = A /\ v = B ) -> <. ( w +R u ) , ( v +R f ) >. = <. ( A +R u ) , ( B +R f ) >. )
6 oveq2 
 |-  ( u = C -> ( A +R u ) = ( A +R C ) )
7 oveq2 
 |-  ( f = D -> ( B +R f ) = ( B +R D ) )
8 opeq12 
 |-  ( ( ( 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 
 |-  ( ( u = C /\ f = D ) -> <. ( A +R u ) , ( B +R f ) >. = <. ( A +R C ) , ( B +R D ) >. )
10 5 9 sylan9eq 
 |-  ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( w +R u ) , ( v +R f ) >. = <. ( A +R C ) , ( B +R D ) >. )
11 df-add 
 |-  + = { <. <. 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 
 |-  CC = ( R. X. R. )
13 12 eleq2i 
 |-  ( x e. CC <-> x e. ( R. X. R. ) )
14 12 eleq2i 
 |-  ( y e. CC <-> y e. ( R. X. R. ) )
15 13 14 anbi12i 
 |-  ( ( x e. CC /\ y e. CC ) <-> ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) )
16 15 anbi1i 
 |-  ( ( ( 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 
 |-  { <. <. 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 
 |-  + = { <. <. 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 
 |-  ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. + <. C , D >. ) = <. ( A +R C ) , ( B +R D ) >. )