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Theorem addsrpr 9496
Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
addsrpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  +R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  )

Proof of Theorem addsrpr
Dummy variables  x  y  z  w  v  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 4865 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 9488 . . . . 5  |-  ~R  e.  _V
32ecelqsi 7416 . . . 4  |-  ( <. A ,  B >.  e.  ( P.  X.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
41, 3syl 17 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
5 opelxpi 4865 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 7416 . . . 4  |-  ( <. C ,  D >.  e.  ( P.  X.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
75, 6syl 17 . . 3  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
84, 7anim12i 569 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) ) )
9 eqid 2450 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2450 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 457 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2450 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R
13 opeq12 4167 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 7397 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2460 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 710 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) 
<->  ( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) ) )
17 simpl 459 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 6303 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  +P.  C
)  =  ( A  +P.  C ) )
19 simpr 463 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 6303 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  +P.  D
)  =  ( B  +P.  D ) )
2118, 20opeq12d 4173 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +P.  C
) ,  ( v  +P.  D ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
2221eceq1d 7397 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
2322eqeq2d 2460 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  <->  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
)
2416, 23anbi12d 716 . . . . 5  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P. 
D ) >. ]  ~R  ) 
<->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  ) ) )
2524spc2egv 3135 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  E. w E. v
( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  ) ) )
26 opeq12 4167 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2726eceq1d 7397 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
2827eqeq2d 2460 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
2928anbi2d 709 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) 
<->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) ) )
30 simpl 459 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3130oveq2d 6304 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  +P.  u
)  =  ( w  +P.  C ) )
32 simpr 463 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3332oveq2d 6304 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  +P.  t
)  =  ( v  +P.  D ) )
3431, 33opeq12d 4173 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( w  +P.  C ) ,  ( v  +P.  D ) >.
)
3534eceq1d 7397 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( w  +P.  C
) ,  ( v  +P.  D ) >. ]  ~R  )
3635eqeq2d 2460 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  <->  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  ) )
3729, 36anbi12d 716 . . . . . 6  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) 
<->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  ) ) )
3837spc2egv 3135 . . . . 5  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P. 
D ) >. ]  ~R  )  ->  E. u E. t
( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  ) ) )
39382eximdv 1765 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( E. w E. v ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  C ) ,  ( v  +P. 
D ) >. ]  ~R  )  ->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  ) ) )
4025, 39sylan9 662 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( (
( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) )
4111, 12, 40mp2ani 683 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
42 ecexg 7364 . . . 4  |-  (  ~R  e.  _V  ->  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  e.  _V )
432, 42ax-mp 5 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  e.  _V
44 simp1 1007 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
4544eqeq1d 2452 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  ( x  =  [ <. w ,  v >. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  ) )
46 simp2 1008 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  y  =  [ <. C ,  D >. ]  ~R  )
4746eqeq1d 2452 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  ( y  =  [ <. u ,  t >. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) )
4845, 47anbi12d 716 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  ( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  <->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) ) )
49 simp3 1009 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
5049eqeq1d 2452 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  ( z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  <->  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
5148, 50anbi12d 716 . . . . 5  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  ( ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  )  /\  [
<. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) )
52514exbidv 1771 . . . 4  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  ( E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) 
<->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  ) ) )
53 addsrmo 9494 . . . 4  |-  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
54 df-plr 9479 . . . . 5  |-  +R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  ) ) }
55 df-nr 9478 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5655eleq2i 2520 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5755eleq2i 2520 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5856, 57anbi12i 702 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
5958anbi1i 700 . . . . . 6  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  <->  ( (
x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  ) ) )
6059oprabbii 6343 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  (
( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) }
6154, 60eqtri 2472 . . . 4  |-  +R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  (
( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) }
6252, 53, 61ovig 6415 . . 3  |-  ( ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [
<. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  e.  _V )  ->  ( E. w E. v E. u E. t
( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  )  -> 
( [ <. A ,  B >. ]  ~R  +R  [
<. C ,  D >. ]  ~R  )  =  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  ) )
6343, 62mp3an3 1352 . 2  |-  ( ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [
<. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  -> 
( E. w E. v E. u E. t
( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  )  -> 
( [ <. A ,  B >. ]  ~R  +R  [
<. C ,  D >. ]  ~R  )  =  [ <. ( A  +P.  C
) ,  ( B  +P.  D ) >. ]  ~R  ) )
648, 41, 63sylc 62 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  +R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 984    = wceq 1443   E.wex 1662    e. wcel 1886   _Vcvv 3044   <.cop 3973    X. cxp 4831  (class class class)co 6288   {coprab 6289   [cec 7358   /.cqs 7359   P.cnp 9281    +P. cpp 9283    ~R cer 9286   R.cnr 9287    +R cplr 9291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-omul 7184  df-er 7360  df-ec 7362  df-qs 7366  df-ni 9294  df-pli 9295  df-mi 9296  df-lti 9297  df-plpq 9330  df-mpq 9331  df-ltpq 9332  df-enq 9333  df-nq 9334  df-erq 9335  df-plq 9336  df-mq 9337  df-1nq 9338  df-rq 9339  df-ltnq 9340  df-np 9403  df-plp 9405  df-ltp 9407  df-enr 9477  df-nr 9478  df-plr 9479
This theorem is referenced by:  addclsr  9504  addcomsr  9508  addasssr  9509  distrsr  9512  m1p1sr  9513  0idsr  9518  ltasr  9521
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