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Theorem mulsrmo 7353
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 7344 . . . . . . . . . . . . . . . 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 7351 . . . . . . . . . . . . . . . 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 7350 . . . . . . . . . . . . . . . . 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 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 )  +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 6354 . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 506 . . . . . . . . . . . 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 508 . . . . . . . . . . . 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 2131 . . . . . . . . . . 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 368 . . . . . . . . . 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 1826 . . . . . . . . 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 1826 . . . . . . . 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 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
)  +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 1826 . . . . . 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 1826 . . . . 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 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
)  +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 1804 . . 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 3632 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  -> 
<. w ,  v >.  =  <. s ,  f
>. )
2221eceq1d 6344 . . . . . . . . . 10  |-  ( ( w  =  s  /\  v  =  f )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. s ,  f >. ]  ~R  )
2322eqeq2d 2100 . . . . . . . . 9  |-  ( ( w  =  s  /\  v  =  f )  ->  ( A  =  [ <. w ,  v >. ]  ~R  <->  A  =  [ <. s ,  f >. ]  ~R  ) )
2423anbi1d 454 . . . . . . . 8  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( A  =  [ <. w ,  v
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  ) ) )
25 simpl 108 . . . . . . . . . . . . 13  |-  ( ( w  =  s  /\  v  =  f )  ->  w  =  s )
2625oveq1d 5683 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  .P.  u
)  =  ( s  .P.  u ) )
27 simpr 109 . . . . . . . . . . . . 13  |-  ( ( w  =  s  /\  v  =  f )  ->  v  =  f )
2827oveq1d 5683 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  .P.  t
)  =  ( f  .P.  t ) )
2926, 28oveq12d 5686 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( w  .P.  u )  +P.  (
v  .P.  t )
)  =  ( ( s  .P.  u )  +P.  ( f  .P.  t ) ) )
3025oveq1d 5683 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( w  .P.  t
)  =  ( s  .P.  t ) )
3127oveq1d 5683 . . . . . . . . . . . 12  |-  ( ( w  =  s  /\  v  =  f )  ->  ( v  .P.  u
)  =  ( f  .P.  u ) )
3230, 31oveq12d 5686 . . . . . . . . . . 11  |-  ( ( w  =  s  /\  v  =  f )  ->  ( ( w  .P.  t )  +P.  (
v  .P.  u )
)  =  ( ( s  .P.  t )  +P.  ( f  .P.  u ) ) )
3329, 32opeq12d 3638 . . . . . . . . . 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 6344 . . . . . . . . 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 2100 . . . . . . . 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 458 . . . . . . 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 3632 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  -> 
<. u ,  t >.  =  <. g ,  h >. )
3837eceq1d 6344 . . . . . . . . . 10  |-  ( ( u  =  g  /\  t  =  h )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. g ,  h >. ]  ~R  )
3938eqeq2d 2100 . . . . . . . . 9  |-  ( ( u  =  g  /\  t  =  h )  ->  ( B  =  [ <. u ,  t >. ]  ~R  <->  B  =  [ <. g ,  h >. ]  ~R  ) )
4039anbi2d 453 . . . . . . . 8  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. u ,  t
>. ]  ~R  )  <->  ( A  =  [ <. s ,  f
>. ]  ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )
41 simpl 108 . . . . . . . . . . . . 13  |-  ( ( u  =  g  /\  t  =  h )  ->  u  =  g )
4241oveq2d 5684 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  .P.  u
)  =  ( s  .P.  g ) )
43 simpr 109 . . . . . . . . . . . . 13  |-  ( ( u  =  g  /\  t  =  h )  ->  t  =  h )
4443oveq2d 5684 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  .P.  t
)  =  ( f  .P.  h ) )
4542, 44oveq12d 5686 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( s  .P.  u )  +P.  (
f  .P.  t )
)  =  ( ( s  .P.  g )  +P.  ( f  .P.  h ) ) )
4643oveq2d 5684 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( s  .P.  t
)  =  ( s  .P.  h ) )
4741oveq2d 5684 . . . . . . . . . . . 12  |-  ( ( u  =  g  /\  t  =  h )  ->  ( f  .P.  u
)  =  ( f  .P.  g ) )
4846, 47oveq12d 5686 . . . . . . . . . . 11  |-  ( ( u  =  g  /\  t  =  h )  ->  ( ( s  .P.  t )  +P.  (
f  .P.  u )
)  =  ( ( s  .P.  h )  +P.  ( f  .P.  g ) ) )
4945, 48opeq12d 3638 . . . . . . . . . 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 6344 . . . . . . . . 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 2100 . . . . . . . 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 458 . . . . . . 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 1854 . . . . . 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 446 . . . . 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 237 . . . 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 1406 . . 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 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
)  +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 2095 . . . . 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 453 . . . 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 1799 . . 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 2010 . 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 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
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1288    = wceq 1290   E.wex 1427    e. wcel 1439   E*wmo 1950   <.cop 3455   class class class wbr 3853    X. cxp 4452  (class class class)co 5668    Er wer 6305   [cec 6306   /.cqs 6307   P.cnp 6913    +P. cpp 6915    .P. cmp 6916    ~R cer 6918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-eprel 4127  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-1o 6197  df-2o 6198  df-oadd 6201  df-omul 6202  df-er 6308  df-ec 6310  df-qs 6314  df-ni 6926  df-pli 6927  df-mi 6928  df-lti 6929  df-plpq 6966  df-mpq 6967  df-enq 6969  df-nqqs 6970  df-plqqs 6971  df-mqqs 6972  df-1nqqs 6973  df-rq 6974  df-ltnqqs 6975  df-enq0 7046  df-nq0 7047  df-0nq0 7048  df-plq0 7049  df-mq0 7050  df-inp 7088  df-iplp 7090  df-imp 7091  df-enr 7335
This theorem is referenced by:  mulsrpr  7355
  Copyright terms: Public domain W3C validator