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Theorem addsrmo 7705
Description: There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
Assertion
Ref Expression
addsrmo  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
Distinct variable groups:    t, A, u, v, w, z    t, B, u, v, w, z

Proof of Theorem addsrmo
Dummy variables  f  g  h  q  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 7697 . . . . . . . . . . . . . . . 16  |-  ~R  Er  ( P.  X.  P. )
21a1i 9 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  ~R  Er  ( P.  X.  P. ) )
3 prsrlem1 7704 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  ( (
( ( w  e. 
P.  /\  v  e.  P. )  /\  (
s  e.  P.  /\  f  e.  P. )
)  /\  ( (
u  e.  P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. )
) )  /\  (
( w  +P.  f
)  =  ( v  +P.  s )  /\  ( u  +P.  h )  =  ( t  +P.  g ) ) ) )
4 addcmpblnr 7701 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( w  e. 
P.  /\  v  e.  P. )  /\  (
s  e.  P.  /\  f  e.  P. )
)  /\  ( (
u  e.  P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. )
) )  ->  (
( ( w  +P.  f )  =  ( v  +P.  s )  /\  ( u  +P.  h )  =  ( t  +P.  g ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  t
) >.  ~R  <. ( s  +P.  g ) ,  ( f  +P.  h
) >. ) )
54imp 123 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( w  e.  P.  /\  v  e.  P. )  /\  (
s  e.  P.  /\  f  e.  P. )
)  /\  ( (
u  e.  P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. )
) )  /\  (
( w  +P.  f
)  =  ( v  +P.  s )  /\  ( u  +P.  h )  =  ( t  +P.  g ) ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  t )
>.  ~R  <. ( s  +P.  g ) ,  ( f  +P.  h )
>. )
63, 5syl 14 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  <. ( w  +P.  u ) ,  ( v  +P.  t
) >.  ~R  <. ( s  +P.  g ) ,  ( f  +P.  h
) >. )
72, 6erthi 6559 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  =  [ <. ( s  +P.  g ) ,  ( f  +P.  h )
>. ]  ~R  )
87adantrlr 482 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  =  [ <. ( s  +P.  g ) ,  ( f  +P.  h )
>. ]  ~R  )
98adantrrr 484 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) ) )  ->  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  =  [ <. ( s  +P.  g ) ,  ( f  +P.  h )
>. ]  ~R  )
10 simprlr 533 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) ) )  ->  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )
11 simprrr 535 . . . . . . . . . . . 12  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) ) )  ->  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  )
129, 10, 113eqtr4d 2213 . . . . . . . . . . 11  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) ) )  ->  z  =  q )
1312expr 373 . . . . . . . . . 10  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  ->  (
( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
s  +P.  g ) ,  ( f  +P.  h ) >. ]  ~R  )  ->  z  =  q ) )
1413exlimdvv 1890 . . . . . . . . 9  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  ->  ( E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
s  +P.  g ) ,  ( f  +P.  h ) >. ]  ~R  )  ->  z  =  q ) )
1514exlimdvv 1890 . . . . . . . 8  |-  ( ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
s  +P.  g ) ,  ( f  +P.  h ) >. ]  ~R  )  ->  z  =  q ) )
1615ex 114 . . . . . . 7  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  ( (
( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  )  ->  z  =  q ) ) )
1716exlimdvv 1890 . . . . . 6  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  ( E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
s  +P.  g ) ,  ( f  +P.  h ) >. ]  ~R  )  ->  z  =  q ) ) )
1817exlimdvv 1890 . . . . 5  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
s  +P.  g ) ,  ( f  +P.  h ) >. ]  ~R  )  ->  z  =  q ) ) )
1918impd 252 . . . 4  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) )  -> 
z  =  q ) )
2019alrimivv 1868 . . 3  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  A. z A. q ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) )  -> 
z  =  q ) )
21 opeq12 3767 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. w ,  v >.  =  <. s ,  f
>. )
2221eceq1d 6549 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. s ,  f >. ]  ~R  )
2322eqeq2d 2182 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  ( A  =  [ <. w ,  v >. ]  ~R  <->  A  =  [ <. s ,  f >. ]  ~R  ) )
2423anbi1d 462 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  ) ) )
25 simpl 108 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  w  =  s )
2625oveq1d 5868 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  +P.  u
)  =  ( s  +P.  u ) )
27 simpr 109 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  v  =  f )
2827oveq1d 5868 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  +P.  t
)  =  ( f  +P.  t ) )
2926, 28opeq12d 3773 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( s  +P.  u ) ,  ( f  +P.  t )
>. )
3029eceq1d 6549 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  )
3130eqeq2d 2182 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( q  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  <->  q  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  ) )
3224, 31anbi12d 470 . . . . . . 7  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )  <->  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  ) ) )
33 opeq12 3767 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. u ,  t >.  =  <. g ,  h >. )
3433eceq1d 6549 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. g ,  h >. ]  ~R  )
3534eqeq2d 2182 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  ( B  =  [ <. u ,  t >. ]  ~R  <->  B  =  [ <. g ,  h >. ]  ~R  ) )
3635anbi2d 461 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )
37 simpl 108 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  u  =  g )
3837oveq2d 5869 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  +P.  u
)  =  ( s  +P.  g ) )
39 simpr 109 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  t  =  h )
4039oveq2d 5869 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  +P.  t
)  =  ( f  +P.  h ) )
4138, 40opeq12d 3773 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. ( s  +P.  u
) ,  ( f  +P.  t ) >.  =  <. ( s  +P.  g ) ,  ( f  +P.  h )
>. )
4241eceq1d 6549 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. ( s  +P.  u ) ,  ( f  +P.  t )
>. ]  ~R  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  )
4342eqeq2d 2182 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( q  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  <->  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) )
4436, 43anbi12d 470 . . . . . . 7  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  )  <->  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) ) )
4532, 44cbvex4v 1923 . . . . . 6  |-  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) 
<->  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
s  +P.  g ) ,  ( f  +P.  h ) >. ]  ~R  ) )
4645anbi2i 454 . . . . 5  |-  ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  <->  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) ) )
4746imbi1i 237 . . . 4  |-  ( ( ( E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  ->  z  =  q )  <->  ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) )  -> 
z  =  q ) )
48472albii 1464 . . 3  |-  ( A. z A. q ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  ->  z  =  q )  <->  A. z A. q ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) )  -> 
z  =  q ) )
4920, 48sylibr 133 . 2  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  A. z A. q ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )  ->  z  =  q ) )
50 eqeq1 2177 . . . . 5  |-  ( z  =  q  ->  (
z  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  <->  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
5150anbi2d 461 . . . 4  |-  ( z  =  q  ->  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) 
<->  ( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) )
52514exbidv 1863 . . 3  |-  ( z  =  q  ->  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) 
<->  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) ) )
5352mo4 2080 . 2  |-  ( E* z E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) 
<-> 
A. z A. q
( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  )  /\  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  ) )  -> 
z  =  q ) )
5449, 53sylibr 133 1  |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1346    = wceq 1348   E.wex 1485   E*wmo 2020    e. wcel 2141   <.cop 3586   class class class wbr 3989    X. cxp 4609  (class class class)co 5853    Er wer 6510   [cec 6511   /.cqs 6512   P.cnp 7253    +P. cpp 7255    ~R cer 7258
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-enr 7688
This theorem is referenced by:  addsrpr  7707
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