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Theorem mulsrmo 7772
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 7763 . . . . . . . . . . . . . . . 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 7770 . . . . . . . . . . . . . . . 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 7769 . . . . . . . . . . . . . . . . 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 124 . . . . . . . . . . . . . . . 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 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 )  +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 6606 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 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 487 . . . . . . . . . . . 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 538 . . . . . . . . . . . 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 540 . . . . . . . . . . . 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 2232 . . . . . . . . . . 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 375 . . . . . . . . . 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 1909 . . . . . . . . 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 1909 . . . . . . . 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 115 . . . . . . 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 1909 . . . . . 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 1909 . . . . 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 254 . . . 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 1886 . . 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 3795 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. w ,  v >.  =  <. s ,  f
>. )
2221eceq1d 6594 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. s ,  f >. ]  ~R  )
2322eqeq2d 2201 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  ( A  =  [ <. w ,  v >. ]  ~R  <->  A  =  [ <. s ,  f >. ]  ~R  ) )
2423anbi1d 465 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  ) ) )
25 simpl 109 . . . . . . . . . . . . 13  |-  ( ( w  =  s  /\  v  =  f )  ->  w  =  s )
2625oveq1d 5910 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  .P.  u
)  =  ( s  .P.  u ) )
27 simpr 110 . . . . . . . . . . . . 13  |-  ( ( w  =  s  /\  v  =  f )  ->  v  =  f )
2827oveq1d 5910 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  .P.  t
)  =  ( f  .P.  t ) )
2926, 28oveq12d 5913 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( w  .P.  u )  +P.  (
v  .P.  t )
)  =  ( ( s  .P.  u )  +P.  ( f  .P.  t ) ) )
3025oveq1d 5910 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  .P.  t
)  =  ( s  .P.  t ) )
3127oveq1d 5910 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  .P.  u
)  =  ( f  .P.  u ) )
3230, 31oveq12d 5913 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( w  .P.  t )  +P.  (
v  .P.  u )
)  =  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) )
3329, 32opeq12d 3801 . . . . . . . . . 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 6594 . . . . . . . . 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 2201 . . . . . . . 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 473 . . . . . . 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 3795 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. u ,  t >.  =  <. g ,  h >. )
3837eceq1d 6594 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. g ,  h >. ]  ~R  )
3938eqeq2d 2201 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  ( B  =  [ <. u ,  t >. ]  ~R  <->  B  =  [ <. g ,  h >. ]  ~R  ) )
4039anbi2d 464 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )
41 simpl 109 . . . . . . . . . . . . 13  |-  ( ( u  =  g  /\  t  =  h )  ->  u  =  g )
4241oveq2d 5911 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  .P.  u
)  =  ( s  .P.  g ) )
43 simpr 110 . . . . . . . . . . . . 13  |-  ( ( u  =  g  /\  t  =  h )  ->  t  =  h )
4443oveq2d 5911 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  .P.  t
)  =  ( f  .P.  h ) )
4542, 44oveq12d 5913 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( s  .P.  u )  +P.  (
f  .P.  t )
)  =  ( ( s  .P.  g )  +P.  ( f  .P.  h ) ) )
4643oveq2d 5911 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  .P.  t
)  =  ( s  .P.  h ) )
4741oveq2d 5911 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  .P.  u
)  =  ( f  .P.  g ) )
4846, 47oveq12d 5913 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( s  .P.  t )  +P.  (
f  .P.  u )
)  =  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) )
4945, 48opeq12d 3801 . . . . . . . . . 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 6594 . . . . . . . . 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 2201 . . . . . . . 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 473 . . . . . . 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 1942 . . . . . 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 457 . . . . 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 238 . . . 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 1482 . . 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 134 . 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 2196 . . . . 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 464 . . . 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 1881 . . 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 2099 . 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 134 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 set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362    = wceq 1364   E.wex 1503   E*wmo 2039    e. wcel 2160   <.cop 3610   class class class wbr 4018    X. cxp 4642  (class class class)co 5895    Er wer 6555   [cec 6556   /.cqs 6557   P.cnp 7319    +P. cpp 7321    .P. cmp 7322    ~R cer 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-recs 6329  df-irdg 6394  df-1o 6440  df-2o 6441  df-oadd 6444  df-omul 6445  df-er 6558  df-ec 6560  df-qs 6564  df-ni 7332  df-pli 7333  df-mi 7334  df-lti 7335  df-plpq 7372  df-mpq 7373  df-enq 7375  df-nqqs 7376  df-plqqs 7377  df-mqqs 7378  df-1nqqs 7379  df-rq 7380  df-ltnqqs 7381  df-enq0 7452  df-nq0 7453  df-0nq0 7454  df-plq0 7455  df-mq0 7456  df-inp 7494  df-iplp 7496  df-imp 7497  df-enr 7754
This theorem is referenced by:  mulsrpr  7774
  Copyright terms: Public domain W3C validator