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Theorem mulcnsr 9957
Description: Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
Assertion
Ref Expression
mulcnsr |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. x. <. C , D >. ) = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. )
Proof of Theorem mulcnsr
Dummy variables x y z w v u f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4932 . 2 |- <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. e. _V
2 oveq1 6657 . . . . 5 |- ( w = A -> ( w .R u ) = ( A .R u ) )
3 oveq1 6657 . . . . . 6 |- ( v = B -> ( v .R f ) = ( B .R f ) )
43oveq2d 6666 . . . . 5 |- ( v = B -> ( -1R .R ( v .R f ) ) = ( -1R .R ( B .R f ) ) )
52, 4oveqan12d 6669 . . . 4 |- ( ( w = A /\ v = B ) -> ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) = ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) )
6 oveq1 6657 . . . . 5 |- ( v = B -> ( v .R u ) = ( B .R u ) )
7 oveq1 6657 . . . . 5 |- ( w = A -> ( w .R f ) = ( A .R f ) )
86, 7oveqan12rd 6670 . . . 4 |- ( ( w = A /\ v = B ) -> ( ( v .R u ) +R ( w .R f ) ) = ( ( B .R u ) +R ( A .R f ) ) )
95, 8opeq12d 4410 . . 3 |- ( ( w = A /\ v = B ) -> <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. = <. ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) , ( ( B .R u ) +R ( A .R f ) ) >. )
10 oveq2 6658 . . . . 5 |- ( u = C -> ( A .R u ) = ( A .R C ) )
11 oveq2 6658 . . . . . 6 |- ( f = D -> ( B .R f ) = ( B .R D ) )
1211oveq2d 6666 . . . . 5 |- ( f = D -> ( -1R .R ( B .R f ) ) = ( -1R .R ( B .R D ) ) )
1310, 12oveqan12d 6669 . . . 4 |- ( ( u = C /\ f = D ) -> ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) = ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) )
14 oveq2 6658 . . . . 5 |- ( u = C -> ( B .R u ) = ( B .R C ) )
15 oveq2 6658 . . . . 5 |- ( f = D -> ( A .R f ) = ( A .R D ) )
1614, 15oveqan12d 6669 . . . 4 |- ( ( u = C /\ f = D ) -> ( ( B .R u ) +R ( A .R f ) ) = ( ( B .R C ) +R ( A .R D ) ) )
1713, 16opeq12d 4410 . . 3 |- ( ( u = C /\ f = D ) -> <. ( ( A .R u ) +R ( -1R .R ( B .R f ) ) ) , ( ( B .R u ) +R ( A .R f ) ) >. = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. )
189, 17sylan9eq 2676 . 2 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> <. ( ( w .R u ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. )
19 df-mul 9948 . . 3 |- x. = { <. <. 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 ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
20 df-c 9942 . . . . . . 7 |- CC = ( R. X. R. )
2120eleq2i 2693 . . . . . 6 |- ( x e. CC <-> x e. ( R. X. R. ) )
2220eleq2i 2693 . . . . . 6 |- ( y e. CC <-> y e. ( R. X. R. ) )
2321, 22anbi12i 733 . . . . 5 |- ( ( x e. CC /\ y e. CC ) <-> ( x e. ( R. X. R. ) /\ y e. ( R. X. R. ) ) )
2423anbi1i 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 ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .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 ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) )
2524oprabbii 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 ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .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 ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
2619, 25eqtri 2644 . 2 |- x. = { <. <. 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 ) +R ( -1R .R ( v .R f ) ) ) , ( ( v .R u ) +R ( w .R f ) ) >. ) ) }
271, 18, 26ov3 6797 1 |- ( ( ( A e. R. /\ B e. R. ) /\ ( C e. R. /\ D e. R. ) ) -> ( <. A , B >. x. <. C , D >. ) = <. ( ( A .R C ) +R ( -1R .R ( B .R D ) ) ) , ( ( B .R C ) +R ( A .R D ) ) >. )
Colors of variables: wff setvar class
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   -1Rcm1r 9690   +R cplr 9691   .R cmr 9692   CCcc 9934   x. cmul 9941
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-mul 9948
This theorem is referenced by:  mulresr  9960  mulcnsrec  9965  axmulf  9967  axi2m1  9980  axcnre  9985
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