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Theorem addsrmo 7663
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 7655 . . . . . . . . . . . . . . . 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 7662 . . . . . . . . . . . . . . . 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 7659 . . . . . . . . . . . . . . . . 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 6526 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 528 . . . . . . . . . . . 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 530 . . . . . . . . . . . 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 2200 . . . . . . . . . . 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 1877 . . . . . . . . 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 1877 . . . . . . . 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 1877 . . . . . 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 1877 . . . . 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 1855 . . 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 3743 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. w ,  v >.  =  <. s ,  f
>. )
2221eceq1d 6516 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. s ,  f >. ]  ~R  )
2322eqeq2d 2169 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  ( A  =  [ <. w ,  v >. ]  ~R  <->  A  =  [ <. s ,  f >. ]  ~R  ) )
2423anbi1d 461 . . . . . . . 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 5839 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  +P.  u
)  =  ( s  +P.  u ) )
27 simpr 109 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  v  =  f )
2827oveq1d 5839 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  +P.  t
)  =  ( f  +P.  t ) )
2926, 28opeq12d 3749 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. ( w  +P.  u
) ,  ( v  +P.  t ) >.  =  <. ( s  +P.  u ) ,  ( f  +P.  t )
>. )
3029eceq1d 6516 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. ( w  +P.  u ) ,  ( v  +P.  t )
>. ]  ~R  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  )
3130eqeq2d 2169 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( q  =  [ <. ( w  +P.  u
) ,  ( v  +P.  t ) >. ]  ~R  <->  q  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  ) )
3224, 31anbi12d 465 . . . . . . 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 3743 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. u ,  t >.  =  <. g ,  h >. )
3433eceq1d 6516 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. g ,  h >. ]  ~R  )
3534eqeq2d 2169 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  ( B  =  [ <. u ,  t >. ]  ~R  <->  B  =  [ <. g ,  h >. ]  ~R  ) )
3635anbi2d 460 . . . . . . . 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 5840 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  +P.  u
)  =  ( s  +P.  g ) )
39 simpr 109 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  t  =  h )
4039oveq2d 5840 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  +P.  t
)  =  ( f  +P.  h ) )
4138, 40opeq12d 3749 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. ( s  +P.  u
) ,  ( f  +P.  t ) >.  =  <. ( s  +P.  g ) ,  ( f  +P.  h )
>. )
4241eceq1d 6516 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. ( s  +P.  u ) ,  ( f  +P.  t )
>. ]  ~R  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  )
4342eqeq2d 2169 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( q  =  [ <. ( s  +P.  u
) ,  ( f  +P.  t ) >. ]  ~R  <->  q  =  [ <. ( s  +P.  g
) ,  ( f  +P.  h ) >. ]  ~R  ) )
4436, 43anbi12d 465 . . . . . . 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 1910 . . . . . 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 453 . . . . 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 1451 . . 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 2164 . . . . 5  |-  ( z  =  q  ->  (
z  =  [ <. ( w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  <->  q  =  [ <. (
w  +P.  u ) ,  ( v  +P.  t ) >. ]  ~R  ) )
5150anbi2d 460 . . . 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 1850 . . 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 2067 . 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 1333    = wceq 1335   E.wex 1472   E*wmo 2007    e. wcel 2128   <.cop 3563   class class class wbr 3965    X. cxp 4584  (class class class)co 5824    Er wer 6477   [cec 6478   /.cqs 6479   P.cnp 7211    +P. cpp 7213    ~R cer 7216
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
This theorem is referenced by:  addsrpr  7665
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