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Theorem mulsrmo 9516
Description: There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
Assertion
Ref Expression
mulsrmo  |-  ( ( 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
Distinct variable groups:    t, A, u, v, w, z    t, B, u, v, w, z

Proof of Theorem mulsrmo
Dummy variables  f 
g  h  q  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 9507 . . . . . . . . . . . . . . . 16  |-  ~R  Er  ( P.  X.  P. )
21a1i 11 . . . . . . . . . . . . . . 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 9514 . . . . . . . . . . . . . . . 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 mulcmpblnr 9513 . . . . . . . . . . . . . . . . 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 )  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >.  ~R  <. ( ( s  .P.  g )  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ) )
54imp 436 . . . . . . . . . . . . . . . 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 )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >.  ~R  <. ( ( s  .P.  g )  +P.  ( f  .P.  h
) ) ,  ( ( s  .P.  h
)  +P.  ( f  .P.  g ) ) >.
)
63, 5syl 17 . . . . . . . . . . . . . . 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 )  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >.  ~R  <. ( ( s  .P.  g )  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. )
72, 6erthi 7428 . . . . . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  =  [ <. ( ( s  .P.  g )  +P.  ( f  .P.  h
) ) ,  ( ( s  .P.  h
)  +P.  ( f  .P.  g ) ) >. ]  ~R  )
87adantrlr 737 . . . . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  =  [ <. ( ( s  .P.  g )  +P.  ( f  .P.  h
) ) ,  ( ( s  .P.  h
)  +P.  ( f  .P.  g ) ) >. ]  ~R  )
98adantrrr 739 . . . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
) )  ->  [ <. ( ( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  =  [ <. ( ( s  .P.  g )  +P.  ( f  .P.  h
) ) ,  ( ( s  .P.  h
)  +P.  ( f  .P.  g ) ) >. ]  ~R  )
10 simprlr 781 . . . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
) )  ->  z  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  )
11 simprrr 783 . . . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
) )  ->  q  =  [ <. ( ( s  .P.  g )  +P.  ( f  .P.  h
) ) ,  ( ( s  .P.  h
)  +P.  ( f  .P.  g ) ) >. ]  ~R  )
129, 10, 113eqtr4d 2515 . . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
) )  ->  z  =  q )
1312expr 626 . . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  ->  ( (
( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )  ->  z  =  q ) )
1413exlimdvv 1788 . . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  ->  ( E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  g )  +P.  (
f  .P.  h )
) ,  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) >. ]  ~R  )  ->  z  =  q ) )
1514exlimdvv 1788 . . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  g )  +P.  (
f  .P.  h )
) ,  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) >. ]  ~R  )  ->  z  =  q ) )
1615ex 441 . . . . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  g )  +P.  (
f  .P.  h )
) ,  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) >. ]  ~R  )  ->  z  =  q ) ) )
1716exlimdvv 1788 . . . . . 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 )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )  ->  z  =  q ) ) )
1817exlimdvv 1788 . . . . 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 )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  ->  ( E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )  ->  z  =  q ) ) )
1918impd 438 . . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
)  ->  z  =  q ) )
2019alrimivv 1782 . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
)  ->  z  =  q ) )
21 opeq12 4160 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. w ,  v >.  =  <. s ,  f
>. )
2221eceq1d 7418 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. s ,  f >. ]  ~R  )
2322eqeq2d 2481 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  ( A  =  [ <. w ,  v >. ]  ~R  <->  A  =  [ <. s ,  f >. ]  ~R  ) )
2423anbi1d 719 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  ) ) )
25 simpl 464 . . . . . . . . . . . . 13  |-  ( ( w  =  s  /\  v  =  f )  ->  w  =  s )
2625oveq1d 6323 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  .P.  u
)  =  ( s  .P.  u ) )
27 simpr 468 . . . . . . . . . . . . 13  |-  ( ( w  =  s  /\  v  =  f )  ->  v  =  f )
2827oveq1d 6323 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  .P.  t
)  =  ( f  .P.  t ) )
2926, 28oveq12d 6326 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( w  .P.  u )  +P.  (
v  .P.  t )
)  =  ( ( s  .P.  u )  +P.  ( f  .P.  t ) ) )
3025oveq1d 6323 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  .P.  t
)  =  ( s  .P.  t ) )
3127oveq1d 6323 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  .P.  u
)  =  ( f  .P.  u ) )
3230, 31oveq12d 6326 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( w  .P.  t )  +P.  (
v  .P.  u )
)  =  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) )
3329, 32opeq12d 4166 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( s  .P.  u )  +P.  ( f  .P.  t
) ) ,  ( ( s  .P.  t
)  +P.  ( f  .P.  u ) ) >.
)
3433eceq1d 7418 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  =  [ <. ( ( s  .P.  u
)  +P.  ( f  .P.  t ) ) ,  ( ( s  .P.  t )  +P.  (
f  .P.  u )
) >. ]  ~R  )
3534eqeq2d 2481 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( q  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  <->  q  =  [ <. ( ( s  .P.  u )  +P.  (
f  .P.  t )
) ,  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) >. ]  ~R  ) )
3624, 35anbi12d 725 . . . . . . 7  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  <->  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  u )  +P.  (
f  .P.  t )
) ,  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) >. ]  ~R  ) ) )
37 opeq12 4160 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. u ,  t >.  =  <. g ,  h >. )
3837eceq1d 7418 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. g ,  h >. ]  ~R  )
3938eqeq2d 2481 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  ( B  =  [ <. u ,  t >. ]  ~R  <->  B  =  [ <. g ,  h >. ]  ~R  ) )
4039anbi2d 718 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )
41 simpl 464 . . . . . . . . . . . . 13  |-  ( ( u  =  g  /\  t  =  h )  ->  u  =  g )
4241oveq2d 6324 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  .P.  u
)  =  ( s  .P.  g ) )
43 simpr 468 . . . . . . . . . . . . 13  |-  ( ( u  =  g  /\  t  =  h )  ->  t  =  h )
4443oveq2d 6324 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  .P.  t
)  =  ( f  .P.  h ) )
4542, 44oveq12d 6326 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( s  .P.  u )  +P.  (
f  .P.  t )
)  =  ( ( s  .P.  g )  +P.  ( f  .P.  h ) ) )
4643oveq2d 6324 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  .P.  t
)  =  ( s  .P.  h ) )
4741oveq2d 6324 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  .P.  u
)  =  ( f  .P.  g ) )
4846, 47oveq12d 6326 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( s  .P.  t )  +P.  (
f  .P.  u )
)  =  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) )
4945, 48opeq12d 4166 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. ( ( s  .P.  u )  +P.  (
f  .P.  t )
) ,  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) >.  =  <. ( ( s  .P.  g )  +P.  ( f  .P.  h
) ) ,  ( ( s  .P.  h
)  +P.  ( f  .P.  g ) ) >.
)
5049eceq1d 7418 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. ( ( s  .P.  u )  +P.  ( f  .P.  t
) ) ,  ( ( s  .P.  t
)  +P.  ( f  .P.  u ) ) >. ]  ~R  =  [ <. ( ( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
5150eqeq2d 2481 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( q  =  [ <. ( ( s  .P.  u )  +P.  (
f  .P.  t )
) ,  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) >. ]  ~R  <->  q  =  [ <. ( ( s  .P.  g )  +P.  (
f  .P.  h )
) ,  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) >. ]  ~R  ) )
5240, 51anbi12d 725 . . . . . . 7  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  u )  +P.  (
f  .P.  t )
) ,  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) >. ]  ~R  )  <->  ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  g )  +P.  (
f  .P.  h )
) ,  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) >. ]  ~R  ) ) )
5336, 52cbvex4v 2139 . . . . . 6  |-  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  <->  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
)
5453anbi2i 708 . . . . 5  |-  ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  <->  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. ( ( s  .P.  g )  +P.  (
f  .P.  h )
) ,  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) >. ]  ~R  ) ) )
5554imbi1i 332 . . . 4  |-  ( ( ( E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  ->  z  =  q )  <->  ( ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
)  ->  z  =  q ) )
56552albii 1700 . . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. s E. f E. g E. h ( ( A  =  [ <. s ,  f >. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  )  /\  q  =  [ <. (
( s  .P.  g
)  +P.  ( f  .P.  h ) ) ,  ( ( s  .P.  h )  +P.  (
f  .P.  g )
) >. ]  ~R  )
)  ->  z  =  q ) )
5720, 56sylibr 217 . 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  ->  z  =  q ) )
58 eqeq1 2475 . . . . 5  |-  ( z  =  q  ->  (
z  =  [ <. ( ( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  <->  q  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) )
5958anbi2d 718 . . . 4  |-  ( z  =  q  ->  (
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  <->  ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) )
60594exbidv 1780 . . 3  |-  ( z  =  q  ->  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  <->  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) )
6160mo4 2366 . 2  |-  ( E* z E. w E. v E. u E. t
( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  /\  E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  /\  q  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  ->  z  =  q ) )
6257, 61sylibr 217 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   E*wmo 2320   <.cop 3965   class class class wbr 4395    X. cxp 4837  (class class class)co 6308    Er wer 7378   [cec 7379   /.cqs 7380   P.cnp 9302    +P. cpp 9304    .P. cmp 9305    ~R cer 9307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ec 7383  df-qs 7387  df-ni 9315  df-pli 9316  df-mi 9317  df-lti 9318  df-plpq 9351  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-plq 9357  df-mq 9358  df-1nq 9359  df-rq 9360  df-ltnq 9361  df-np 9424  df-plp 9426  df-mp 9427  df-ltp 9428  df-enr 9498
This theorem is referenced by:  mulsrpr  9518
  Copyright terms: Public domain W3C validator