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Theorem addsrpr 7270
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 4459 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 7262 . . . . 5  |-  ~R  e.  _V
32ecelqsi 6326 . . . 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 4459 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 6326 . . . 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 331 . 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 2088 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2088 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 266 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2088 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R
13 opeq12 3619 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 6308 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2099 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 453 . . . . . 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 107 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 5649 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  +P.  C
)  =  ( A  +P.  C ) )
19 simpr 108 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 5649 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  +P.  D
)  =  ( B  +P.  D ) )
2118, 20opeq12d 3625 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +P.  C
) ,  ( v  +P.  D ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
2221eceq1d 6308 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
2322eqeq2d 2099 . . . . . 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 457 . . . . 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 2708 . . . 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 3619 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2726eceq1d 6308 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
2827eqeq2d 2099 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
2928anbi2d 452 . . . . . . 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 107 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3130oveq2d 5650 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  +P.  u
)  =  ( w  +P.  C ) )
32 simpr 108 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3332oveq2d 5650 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  +P.  t
)  =  ( v  +P.  D ) )
3431, 33opeq12d 3625 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( w  +P.  C ) ,  ( v  +P.  D ) >.
)
3534eceq1d 6308 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( w  +P.  C
) ,  ( v  +P.  D ) >. ]  ~R  )
3635eqeq2d 2099 . . . . . . 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 457 . . . . . 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 2708 . . . . 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 1810 . . . 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 401 . . 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 423 . 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 6276 . . . 4  |-  (  ~R  e.  _V  ->  [ <. ( A  +P.  C ) ,  ( B  +P.  D
) >. ]  ~R  e.  _V )
432, 42ax-mp 7 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  e.  _V
44 simp1 943 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
4544eqeq1d 2096 . . . . . . 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 944 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  y  =  [ <. C ,  D >. ]  ~R  )
4746eqeq1d 2096 . . . . . . 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 457 . . . . . 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 945 . . . . . . 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 2096 . . . . . 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 457 . . . . 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 1798 . . . 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 7268 . . . 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 7253 . . . . 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 7252 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5655eleq2i 2154 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5755eleq2i 2154 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5856, 57anbi12i 448 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
5958anbi1i 446 . . . . . 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 5686 . . . . 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 2108 . . . 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 5748 . . 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 1262 . 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 61 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 102    /\ w3a 924    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   <.cop 3444    X. cxp 4426  (class class class)co 5634   {coprab 5635   [cec 6270   /.cqs 6271   P.cnp 6829    +P. cpp 6831    ~R cer 6834   R.cnr 6835    +R cplr 6839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-enr 7251  df-nr 7252  df-plr 7253
This theorem is referenced by:  addclsr  7278  addcomsrg  7280  addasssrg  7281  distrsrg  7284  m1p1sr  7285  0idsr  7292  ltasrg  7295  prsradd  7310  pitonnlem2  7363
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