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Theorem addcnsr 9559
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.
StepHypRef Expression
1 opex 4681 . 2  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
2 oveq1 6308 . . . 4  |-  ( w  =  A  ->  (
w  +R  u )  =  ( A  +R  u ) )
3 oveq1 6308 . . . 4  |-  ( v  =  B  ->  (
v  +R  f )  =  ( B  +R  f ) )
4 opeq12 4186 . . . 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
) >. )
52, 3, 4syl2an 479 . . 3  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +R  u
) ,  ( v  +R  f ) >.  =  <. ( A  +R  u ) ,  ( B  +R  f )
>. )
6 oveq2 6309 . . . 4  |-  ( u  =  C  ->  ( A  +R  u )  =  ( A  +R  C
) )
7 oveq2 6309 . . . 4  |-  ( f  =  D  ->  ( B  +R  f )  =  ( B  +R  D
) )
8 opeq12 4186 . . . 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
) >. )
96, 7, 8syl2an 479 . . 3  |-  ( ( u  =  C  /\  f  =  D )  -> 
<. ( A  +R  u
) ,  ( B  +R  f ) >.  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
105, 9sylan9eq 2483 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  +R  u
) ,  ( v  +R  f ) >.  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
11 df-add 9550 . . 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 9545 . . . . . . 7  |-  CC  =  ( R.  X.  R. )
1312eleq2i 2500 . . . . . 6  |-  ( x  e.  CC  <->  x  e.  ( R.  X.  R. )
)
1412eleq2i 2500 . . . . . 6  |-  ( y  e.  CC  <->  y  e.  ( R.  X.  R. )
)
1513, 14anbi12i 701 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  <->  ( x  e.  ( R. 
X.  R. )  /\  y  e.  ( R.  X.  R. ) ) )
1615anbi1i 699 . . . 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 ) >. )
) )
1716oprabbii 6356 . . 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 ) >. )
) }
1811, 17eqtri 2451 . 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 ) >. )
) }
191, 10, 18ov3 6443 1  |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. )
)  ->  ( <. A ,  B >.  +  <. C ,  D >. )  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1868   <.cop 4002    X. cxp 4847  (class class class)co 6301   {coprab 6302   R.cnr 9290    +R cplr 9294   CCcc 9537    + caddc 9542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-ov 6304  df-oprab 6305  df-c 9545  df-add 9550
This theorem is referenced by:  addresr  9562  addcnsrec  9567  axaddf  9569  axcnre  9588
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