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Theorem addsrpr 7665
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 4618 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 7657 . . . . 5  |-  ~R  e.  _V
32ecelqsi 6534 . . . 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 4618 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 6534 . . . 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 336 . 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 2157 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2157 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 270 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2157 . . 3  |-  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R
13 opeq12 3743 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 6516 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2169 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 461 . . . . . 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 108 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 5839 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  +P.  C
)  =  ( A  +P.  C ) )
19 simpr 109 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 5839 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  +P.  D
)  =  ( B  +P.  D ) )
2118, 20opeq12d 3749 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( w  +P.  C
) ,  ( v  +P.  D ) >.  =  <. ( A  +P.  C ) ,  ( B  +P.  D ) >.
)
2221eceq1d 6516 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( w  +P.  C ) ,  ( v  +P.  D ) >. ]  ~R  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
2322eqeq2d 2169 . . . . . 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 465 . . . . 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 2802 . . . 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 3743 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
2726eceq1d 6516 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
2827eqeq2d 2169 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
2928anbi2d 460 . . . . . . 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 108 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3130oveq2d 5840 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  +P.  u
)  =  ( w  +P.  C ) )
32 simpr 109 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3332oveq2d 5840 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  +P.  t
)  =  ( v  +P.  D ) )
3431, 33opeq12d 3749 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( w  +P.  C ) ,  ( v  +P.  D ) >.
)
3534eceq1d 6516 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( w  +P.  C
) ,  ( v  +P.  D ) >. ]  ~R  )
3635eqeq2d 2169 . . . . . . 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 465 . . . . . 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 2802 . . . . 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 1862 . . . 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 407 . . 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 429 . 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 6484 . . . 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 982 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
4544eqeq1d 2166 . . . . . . 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 983 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )  ->  y  =  [ <. C ,  D >. ]  ~R  )
4746eqeq1d 2166 . . . . . . 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 465 . . . . . 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 984 . . . . . . 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 2166 . . . . . 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 465 . . . . 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 1850 . . . 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 7663 . . . 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 7648 . . . . 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 7647 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
5655eleq2i 2224 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5755eleq2i 2224 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
5856, 57anbi12i 456 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
5958anbi1i 454 . . . . . 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 5876 . . . . 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 2178 . . . 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 5942 . . 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 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 103    /\ w3a 963    = wceq 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712   <.cop 3563    X. cxp 4584  (class class class)co 5824   {coprab 5825   [cec 6478   /.cqs 6479   P.cnp 7211    +P. cpp 7213    ~R cer 7216   R.cnr 7217    +R cplr 7221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-iplp 7388  df-enr 7646  df-nr 7647  df-plr 7648
This theorem is referenced by:  addclsr  7673  addcomsrg  7675  addasssrg  7676  distrsrg  7679  m1p1sr  7680  0idsr  7687  ltasrg  7690  prsradd  7706  pitonnlem2  7767
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