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Theorem addsrpr 9441
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 5023 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 9433 . . . . 5  |-  ~R  e.  _V
32ecelqsi 7357 . . . 4  |-  ( <. A ,  B >.  e.  ( P.  X.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
41, 3syl 16 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
5 opelxpi 5023 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 7357 . . . 4  |-  ( <. C ,  D >.  e.  ( P.  X.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
75, 6syl 16 . . 3  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
84, 7anim12i 566 . 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 2460 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2460 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 455 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2460 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R
13 opeq12 4208 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 7338 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2474 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 704 . . . . . 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 457 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 6290 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  +P.  C
)  =  ( A  +P.  C ) )
19 simpr 461 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 6290 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  +P.  D
)  =  ( B  +P.  D ) )
2118, 20opeq12d 4214 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +P.  C
) ,  ( v  +P.  D ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
2221eceq1d 7338 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
2322eqeq2d 2474 . . . . . 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 710 . . . . 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 3193 . . . 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 4208 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2726eceq1d 7338 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
2827eqeq2d 2474 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
2928anbi2d 703 . . . . . . 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 457 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3130oveq2d 6291 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  +P.  u
)  =  ( w  +P.  C ) )
32 simpr 461 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3332oveq2d 6291 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  +P.  t
)  =  ( v  +P.  D ) )
3431, 33opeq12d 4214 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( w  +P.  C ) ,  ( v  +P.  D ) >.
)
3534eceq1d 7338 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( w  +P.  C
) ,  ( v  +P.  D ) >. ]  ~R  )
3635eqeq2d 2474 . . . . . . 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 710 . . . . . 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 3193 . . . . 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 1683 . . . 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 657 . . 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 678 . 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 7305 . . . 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 991 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
4544eqeq1d 2462 . . . . . . 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 992 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  y  =  [ <. C ,  D >. ]  ~R  )
4746eqeq1d 2462 . . . . . . 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 710 . . . . . 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 993 . . . . . . 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 2462 . . . . . 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 710 . . . . 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 1689 . . . 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 9439 . . . 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 9424 . . . . 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 9423 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5655eleq2i 2538 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5755eleq2i 2538 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5856, 57anbi12i 697 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
5958anbi1i 695 . . . . . 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 6327 . . . . 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 2489 . . . 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 6399 . . 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 1308 . 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 60 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 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3106   <.cop 4026    X. cxp 4990  (class class class)co 6275   {coprab 6276   [cec 7299   /.cqs 7300   P.cnp 9226    +P. cpp 9228    ~R cer 9231   R.cnr 9232    +R cplr 9236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-omul 7125  df-er 7301  df-ec 7303  df-qs 7307  df-ni 9239  df-pli 9240  df-mi 9241  df-lti 9242  df-plpq 9275  df-mpq 9276  df-ltpq 9277  df-enq 9278  df-nq 9279  df-erq 9280  df-plq 9281  df-mq 9282  df-1nq 9283  df-rq 9284  df-ltnq 9285  df-np 9348  df-plp 9350  df-ltp 9352  df-enr 9422  df-nr 9423  df-plr 9424
This theorem is referenced by:  addclsr  9449  addcomsr  9453  addasssr  9454  distrsr  9457  m1p1sr  9458  0idsr  9463  ltasr  9466
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