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Theorem addsrpr 7758
Description: Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
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 4670 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 7750 . . . . 5  |-  ~R  e.  _V
32ecelqsi 6603 . . . 4  |-  ( <. A ,  B >.  e.  ( P.  X.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
41, 3syl 14 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
5 opelxpi 4670 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 6603 . . . 4  |-  ( <. C ,  D >.  e.  ( P.  X.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
75, 6syl 14 . . 3  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
84, 7anim12i 338 . 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 2187 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2187 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 272 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2187 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R
13 opeq12 3792 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 6585 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2199 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 465 . . . . . 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 109 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 5903 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  +P.  C
)  =  ( A  +P.  C ) )
19 simpr 110 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 5903 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  +P.  D
)  =  ( B  +P.  D ) )
2118, 20opeq12d 3798 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +P.  C
) ,  ( v  +P.  D ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
2221eceq1d 6585 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
2322eqeq2d 2199 . . . . . 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 473 . . . . 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 2839 . . . 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 3792 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2726eceq1d 6585 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
2827eqeq2d 2199 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
2928anbi2d 464 . . . . . . 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 109 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3130oveq2d 5904 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  +P.  u
)  =  ( w  +P.  C ) )
32 simpr 110 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3332oveq2d 5904 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  +P.  t
)  =  ( v  +P.  D ) )
3431, 33opeq12d 3798 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( w  +P.  C ) ,  ( v  +P.  D ) >.
)
3534eceq1d 6585 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( w  +P.  C
) ,  ( v  +P.  D ) >. ]  ~R  )
3635eqeq2d 2199 . . . . . . 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 473 . . . . . 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 2839 . . . . 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 1892 . . . 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 409 . . 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 432 . 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 6553 . . . 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 998 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
4544eqeq1d 2196 . . . . . . 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 999 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  y  =  [ <. C ,  D >. ]  ~R  )
4746eqeq1d 2196 . . . . . . 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 473 . . . . . 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 1000 . . . . . . 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 2196 . . . . . 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 473 . . . . 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 1880 . . . 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 7756 . . . 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 7741 . . . . 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 7740 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5655eleq2i 2254 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5755eleq2i 2254 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5856, 57anbi12i 460 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
5958anbi1i 458 . . . . . 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 5943 . . . . 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 2208 . . . 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 6010 . . 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 1336 . 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 set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363   E.wex 1502    e. wcel 2158   _Vcvv 2749   <.cop 3607    X. cxp 4636  (class class class)co 5888   {coprab 5889   [cec 6547   /.cqs 6548   P.cnp 7304    +P. cpp 7306    ~R cer 7309   R.cnr 7310    +R cplr 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-1o 6431  df-2o 6432  df-oadd 6435  df-omul 6436  df-er 6549  df-ec 6551  df-qs 6555  df-ni 7317  df-pli 7318  df-mi 7319  df-lti 7320  df-plpq 7357  df-mpq 7358  df-enq 7360  df-nqqs 7361  df-plqqs 7362  df-mqqs 7363  df-1nqqs 7364  df-rq 7365  df-ltnqqs 7366  df-enq0 7437  df-nq0 7438  df-0nq0 7439  df-plq0 7440  df-mq0 7441  df-inp 7479  df-iplp 7481  df-enr 7739  df-nr 7740  df-plr 7741
This theorem is referenced by:  addclsr  7766  addcomsrg  7768  addasssrg  7769  distrsrg  7772  m1p1sr  7773  0idsr  7780  ltasrg  7783  prsradd  7799  pitonnlem2  7860
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