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Theorem addcnsr 9511
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 4711 . 2  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
2 oveq1 6290 . . . 4  |-  ( w  =  A  ->  (
w  +R  u )  =  ( A  +R  u ) )
3 oveq1 6290 . . . 4  |-  ( v  =  B  ->  (
v  +R  f )  =  ( B  +R  f ) )
4 opeq12 4215 . . . 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 477 . . 3  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +R  u
) ,  ( v  +R  f ) >.  =  <. ( A  +R  u ) ,  ( B  +R  f )
>. )
6 oveq2 6291 . . . 4  |-  ( u  =  C  ->  ( A  +R  u )  =  ( A  +R  C
) )
7 oveq2 6291 . . . 4  |-  ( f  =  D  ->  ( B  +R  f )  =  ( B  +R  D
) )
8 opeq12 4215 . . . 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 477 . . 3  |-  ( ( u  =  C  /\  f  =  D )  -> 
<. ( A  +R  u
) ,  ( B  +R  f ) >.  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
105, 9sylan9eq 2528 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  +R  u
) ,  ( v  +R  f ) >.  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
11 df-add 9502 . . 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 9497 . . . . . . 7  |-  CC  =  ( R.  X.  R. )
1312eleq2i 2545 . . . . . 6  |-  ( x  e.  CC  <->  x  e.  ( R.  X.  R. )
)
1412eleq2i 2545 . . . . . 6  |-  ( y  e.  CC  <->  y  e.  ( R.  X.  R. )
)
1513, 14anbi12i 697 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  <->  ( x  e.  ( R. 
X.  R. )  /\  y  e.  ( R.  X.  R. ) ) )
1615anbi1i 695 . . . 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 6335 . . 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 2496 . 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 6422 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 369    = wceq 1379   E.wex 1596    e. wcel 1767   <.cop 4033    X. cxp 4997  (class class class)co 6283   {coprab 6284   R.cnr 9242    +R cplr 9246   CCcc 9489    + caddc 9494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-oprab 6287  df-c 9497  df-add 9502
This theorem is referenced by:  addresr  9514  addcnsrec  9519  axaddf  9521  axcnre  9540
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