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Theorem prsrlem1 6885
Description: Decomposing signed reals into positive reals. Lemma for addsrpr 6888 and mulsrpr 6889. (Contributed by Jim Kingdon, 30-Dec-2019.)
Assertion
Ref Expression
prsrlem1  |-  ( ( ( 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 ) ) ) )
Distinct variable group:    f, g, h, s, t, u, v, w
Allowed substitution hints:    A( w, v, u, t, f, g, h, s)    B( w, v, u, t, f, g, h, s)

Proof of Theorem prsrlem1
Dummy variables  a  b  c  d  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 enrer 6878 . . . . . 6  |-  ~R  Er  ( P.  X.  P. )
2 erdm 6147 . . . . . 6  |-  (  ~R  Er  ( P.  X.  P. )  ->  dom  ~R  =  ( P.  X.  P. )
)
31, 2ax-mp 7 . . . . 5  |-  dom  ~R  =  ( P.  X.  P. )
4 simprll 497 . . . . . 6  |-  ( ( ( 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  ) ) )  ->  A  =  [ <. w ,  v
>. ]  ~R  )
5 simpll 489 . . . . . 6  |-  ( ( ( 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  ) ) )  ->  A  e.  ( ( P.  X.  P. ) /.  ~R  )
)
64, 5eqeltrrd 2131 . . . . 5  |-  ( ( ( 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 ,  v >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
7 ecelqsdm 6207 . . . . 5  |-  ( ( dom  ~R  =  ( P.  X.  P. )  /\  [ <. w ,  v
>. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  <. w ,  v
>.  e.  ( P.  X.  P. ) )
83, 6, 7sylancr 399 . . . 4  |-  ( ( ( 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 ,  v >.  e.  ( P.  X.  P. ) )
9 opelxp 4402 . . . 4  |-  ( <.
w ,  v >.  e.  ( P.  X.  P. ) 
<->  ( w  e.  P.  /\  v  e.  P. )
)
108, 9sylib 131 . . 3  |-  ( ( ( 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. ) )
11 simprrl 499 . . . . . 6  |-  ( ( ( 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  ) ) )  ->  A  =  [ <. s ,  f
>. ]  ~R  )
1211, 5eqeltrrd 2131 . . . . 5  |-  ( ( ( 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  ) ) )  ->  [ <. s ,  f >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
13 ecelqsdm 6207 . . . . 5  |-  ( ( dom  ~R  =  ( P.  X.  P. )  /\  [ <. s ,  f
>. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  <. s ,  f
>.  e.  ( P.  X.  P. ) )
143, 12, 13sylancr 399 . . . 4  |-  ( ( ( 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  ) ) )  ->  <. s ,  f >.  e.  ( P.  X.  P. ) )
15 opelxp 4402 . . . 4  |-  ( <.
s ,  f >.  e.  ( P.  X.  P. ) 
<->  ( s  e.  P.  /\  f  e.  P. )
)
1614, 15sylib 131 . . 3  |-  ( ( ( 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  ) ) )  ->  ( s  e.  P.  /\  f  e. 
P. ) )
1710, 16jca 294 . 2  |-  ( ( ( 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. )
) )
18 simprlr 498 . . . . . 6  |-  ( ( ( 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  ) ) )  ->  B  =  [ <. u ,  t
>. ]  ~R  )
19 simplr 490 . . . . . 6  |-  ( ( ( 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  ) ) )  ->  B  e.  ( ( P.  X.  P. ) /.  ~R  )
)
2018, 19eqeltrrd 2131 . . . . 5  |-  ( ( ( 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  ) ) )  ->  [ <. u ,  t >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
21 ecelqsdm 6207 . . . . 5  |-  ( ( dom  ~R  =  ( P.  X.  P. )  /\  [ <. u ,  t
>. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  <. u ,  t
>.  e.  ( P.  X.  P. ) )
223, 20, 21sylancr 399 . . . 4  |-  ( ( ( 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  ) ) )  ->  <. u ,  t >.  e.  ( P.  X.  P. ) )
23 opelxp 4402 . . . 4  |-  ( <.
u ,  t >.  e.  ( P.  X.  P. ) 
<->  ( u  e.  P.  /\  t  e.  P. )
)
2422, 23sylib 131 . . 3  |-  ( ( ( 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  ) ) )  ->  ( u  e.  P.  /\  t  e. 
P. ) )
25 simprrr 500 . . . . . 6  |-  ( ( ( 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  ) ) )  ->  B  =  [ <. g ,  h >. ]  ~R  )
2625, 19eqeltrrd 2131 . . . . 5  |-  ( ( ( 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  ) ) )  ->  [ <. g ,  h >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
27 ecelqsdm 6207 . . . . 5  |-  ( ( dom  ~R  =  ( P.  X.  P. )  /\  [ <. g ,  h >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  <. g ,  h >.  e.  ( P.  X.  P. ) )
283, 26, 27sylancr 399 . . . 4  |-  ( ( ( 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  ) ) )  ->  <. g ,  h >.  e.  ( P.  X.  P. ) )
29 opelxp 4402 . . . 4  |-  ( <.
g ,  h >.  e.  ( P.  X.  P. ) 
<->  ( g  e.  P.  /\  h  e.  P. )
)
3028, 29sylib 131 . . 3  |-  ( ( ( 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  ) ) )  ->  ( g  e.  P.  /\  h  e. 
P. ) )
3124, 30jca 294 . 2  |-  ( ( ( 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  ) ) )  ->  ( (
u  e.  P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. )
) )
324, 11eqtr3d 2090 . . . . 5  |-  ( ( ( 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 ,  v >. ]  ~R  =  [ <. s ,  f
>. ]  ~R  )
331a1i 9 . . . . . 6  |-  ( ( ( 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. ) )
3433, 8erth 6181 . . . . 5  |-  ( ( ( 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 ,  v >.  ~R  <. s ,  f >.  <->  [ <. w ,  v >. ]  ~R  =  [ <. s ,  f
>. ]  ~R  ) )
3532, 34mpbird 160 . . . 4  |-  ( ( ( 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 ,  v >.  ~R  <. s ,  f >. )
36 df-enr 6869 . . . . . 6  |-  ~R  =  { <. x ,  y
>.  |  ( (
x  e.  ( P. 
X.  P. )  /\  y  e.  ( P.  X.  P. ) )  /\  E. a E. b E. c E. d ( ( x  =  <. a ,  b
>.  /\  y  =  <. c ,  d >. )  /\  ( a  +P.  d
)  =  ( b  +P.  c ) ) ) }
3736ecopoveq 6232 . . . . 5  |-  ( ( ( w  e.  P.  /\  v  e.  P. )  /\  ( s  e.  P.  /\  f  e.  P. )
)  ->  ( <. w ,  v >.  ~R  <. s ,  f >.  <->  ( w  +P.  f )  =  ( v  +P.  s ) ) )
3810, 16, 37syl2anc 397 . . . 4  |-  ( ( ( 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 ,  v >.  ~R  <. s ,  f >.  <->  ( w  +P.  f )  =  ( v  +P.  s ) ) )
3935, 38mpbid 139 . . 3  |-  ( ( ( 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.  f )  =  ( v  +P.  s ) )
4018, 25eqtr3d 2090 . . . . 5  |-  ( ( ( 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  ) ) )  ->  [ <. u ,  t >. ]  ~R  =  [ <. g ,  h >. ]  ~R  )
4133, 22erth 6181 . . . . 5  |-  ( ( ( 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  ) ) )  ->  ( <. u ,  t >.  ~R  <. g ,  h >.  <->  [ <. u ,  t >. ]  ~R  =  [ <. g ,  h >. ]  ~R  ) )
4240, 41mpbird 160 . . . 4  |-  ( ( ( 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  ) ) )  ->  <. u ,  t >.  ~R  <. g ,  h >. )
4336ecopoveq 6232 . . . . 5  |-  ( ( ( u  e.  P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. )
)  ->  ( <. u ,  t >.  ~R  <. g ,  h >.  <->  ( u  +P.  h )  =  ( t  +P.  g ) ) )
4424, 30, 43syl2anc 397 . . . 4  |-  ( ( ( 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  ) ) )  ->  ( <. u ,  t >.  ~R  <. g ,  h >.  <->  ( u  +P.  h )  =  ( t  +P.  g ) ) )
4542, 44mpbid 139 . . 3  |-  ( ( ( 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  ) ) )  ->  ( u  +P.  h )  =  ( t  +P.  g ) )
4639, 45jca 294 . 2  |-  ( ( ( 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.  f )  =  ( v  +P.  s )  /\  (
u  +P.  h )  =  ( t  +P.  g ) ) )
4717, 31, 46jca31 296 1  |-  ( ( ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   <.cop 3406   class class class wbr 3792    X. cxp 4371   dom cdm 4373  (class class class)co 5540    Er wer 6134   [cec 6135   /.cqs 6136   P.cnp 6447    +P. cpp 6449    ~R cer 6452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-enr 6869
This theorem is referenced by:  addsrmo  6886  mulsrmo  6887
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