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Theorem addcnsr 8773
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 4253 . 2  |-  <. ( A  +R  C ) ,  ( B  +R  D
) >.  e.  _V
2 oveq1 5881 . . . 4  |-  ( w  =  A  ->  (
w  +R  u )  =  ( A  +R  u ) )
3 oveq1 5881 . . . 4  |-  ( v  =  B  ->  (
v  +R  f )  =  ( B  +R  f ) )
4 opeq12 3814 . . . 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 463 . . 3  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +R  u
) ,  ( v  +R  f ) >.  =  <. ( A  +R  u ) ,  ( B  +R  f )
>. )
6 oveq2 5882 . . . 4  |-  ( u  =  C  ->  ( A  +R  u )  =  ( A  +R  C
) )
7 oveq2 5882 . . . 4  |-  ( f  =  D  ->  ( B  +R  f )  =  ( B  +R  D
) )
8 opeq12 3814 . . . 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 463 . . 3  |-  ( ( u  =  C  /\  f  =  D )  -> 
<. ( A  +R  u
) ,  ( B  +R  f ) >.  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
105, 9sylan9eq 2348 . 2  |-  ( ( ( w  =  A  /\  v  =  B )  /\  ( u  =  C  /\  f  =  D ) )  ->  <. ( w  +R  u
) ,  ( v  +R  f ) >.  =  <. ( A  +R  C ) ,  ( B  +R  D )
>. )
11 df-add 8764 . . 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 8759 . . . . . . 7  |-  CC  =  ( R.  X.  R. )
1312eleq2i 2360 . . . . . 6  |-  ( x  e.  CC  <->  x  e.  ( R.  X.  R. )
)
1412eleq2i 2360 . . . . . 6  |-  ( y  e.  CC  <->  y  e.  ( R.  X.  R. )
)
1513, 14anbi12i 678 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  CC )  <->  ( x  e.  ( R. 
X.  R. )  /\  y  e.  ( R.  X.  R. ) ) )
1615anbi1i 676 . . . 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 5919 . . 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 2316 . 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 6000 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 set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656    X. cxp 4703  (class class class)co 5874   {coprab 5875   R.cnr 8505    +R cplr 8509   CCcc 8751    + caddc 8756
This theorem is referenced by:  addresr  8776  addcnsrec  8781  axaddf  8783  axcnre  8802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-c 8759  df-add 8764
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