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Theorem caucvgprlemladdfu 6747
Description: Lemma for caucvgpr 6752. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  |-  ( ph  ->  F : N. --> Q. )
caucvgpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
caucvgpr.bnd  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
caucvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
caucvgprlemladd.s  |-  ( ph  ->  S  e.  Q. )
Assertion
Ref Expression
caucvgprlemladdfu  |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  C_  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
Distinct variable groups:    A, j    j, F, u, l    n, F, k    k, L, j    S, l, u, j    j,
k
Allowed substitution hints:    ph( u, j, k, n, l)    A( u, k, n, l)    S( k, n)    L( u, n, l)

Proof of Theorem caucvgprlemladdfu
Dummy variables  m  r  s  t  v  w  z  f  g  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . . . . 7  |-  ( ph  ->  F : N. --> Q. )
2 caucvgpr.cau . . . . . . 7  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  /\  ( F `  k ) 
<Q  ( ( F `  n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  )
) ) ) )
3 caucvgpr.bnd . . . . . . 7  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
4 caucvgpr.lim . . . . . . 7  |-  L  = 
<. { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >.
51, 2, 3, 4caucvgprlemcl 6746 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 caucvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 6617 . . . . . . 7  |-  ( S  e.  Q.  ->  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  e.  P. )
86, 7syl 14 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )
9 df-iplp 6538 . . . . . . 7  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
10 addclnq 6445 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvu 6583 . . . . . 6  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
125, 8, 11syl2anc 391 . . . . 5  |-  ( ph  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
1312biimpa 280 . . . 4  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  E. s  e.  ( 2nd `  L
) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) r  =  ( s  +Q  t ) )
14 breq2 3765 . . . . . . . . . . . . . . . 16  |-  ( u  =  s  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
1514rexbidv 2324 . . . . . . . . . . . . . . 15  |-  ( u  =  s  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
164fveq2i 5159 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  L )  =  ( 2nd `  <. { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) } ,  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )
17 nqex 6433 . . . . . . . . . . . . . . . . . 18  |-  Q.  e.  _V
1817rabex 3898 . . . . . . . . . . . . . . . . 17  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  _V
1917rabex 3898 . . . . . . . . . . . . . . . . 17  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  _V
2018, 19op2nd 5752 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  <. { l  e. 
Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) } ,  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u } >. )  =  { u  e. 
Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }
2116, 20eqtri 2060 . . . . . . . . . . . . . . 15  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
2215, 21elrab2 2697 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2322biimpi 113 . . . . . . . . . . . . 13  |-  ( s  e.  ( 2nd `  L
)  ->  ( s  e.  Q.  /\  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
s ) )
2423adantr 261 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  ->  (
s  e.  Q.  /\  E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
2524adantl 262 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( s  e.  Q.  /\ 
E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
2625adantr 261 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  e.  Q.  /\ 
E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s ) )
2726simpld 105 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
s  e.  Q. )
28 vex 2557 . . . . . . . . . . . . . 14  |-  t  e. 
_V
29 breq2 3765 . . . . . . . . . . . . . 14  |-  ( u  =  t  ->  ( S  <Q  u  <->  S  <Q  t ) )
30 ltnqex 6619 . . . . . . . . . . . . . . 15  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 6620 . . . . . . . . . . . . . . 15  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op2nd 5752 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { u  |  S  <Q  u }
3328, 29, 32elab2 2687 . . . . . . . . . . . . 13  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  S  <Q  t )
34 ltrelnq 6435 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3534brel 4370 . . . . . . . . . . . . 13  |-  ( S 
<Q  t  ->  ( S  e.  Q.  /\  t  e.  Q. ) )
3633, 35sylbi 114 . . . . . . . . . . . 12  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  ( S  e.  Q.  /\  t  e. 
Q. ) )
3736simprd 107 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  t  e.  Q. )
3837ad2antll 460 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
t  e.  Q. )
3938adantr 261 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
t  e.  Q. )
40 addclnq 6445 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4127, 39, 40syl2anc 391 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  +Q  t
)  e.  Q. )
42 eleq1 2100 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4342adantl 262 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4441, 43mpbird 156 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  Q. )
4526simprd 107 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s )
46 fveq2 5156 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( F `  j )  =  ( F `  m ) )
47 opeq1 3546 . . . . . . . . . . . . . . 15  |-  ( j  =  m  ->  <. j ,  1o >.  =  <. m ,  1o >. )
4847eceq1d 6120 . . . . . . . . . . . . . 14  |-  ( j  =  m  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. m ,  1o >. ]  ~Q  )
4948fveq2d 5160 . . . . . . . . . . . . 13  |-  ( j  =  m  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )
5046, 49oveq12d 5508 . . . . . . . . . . . 12  |-  ( j  =  m  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) ) )
5150breq1d 3771 . . . . . . . . . . 11  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  <->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s ) )
5251cbvrexv 2531 . . . . . . . . . 10  |-  ( E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  s  <->  E. m  e.  N.  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s )
5345, 52sylib 127 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. m  e.  N.  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )
5433biimpi 113 . . . . . . . . . . . . . . . . 17  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  S  <Q  t )
5554ad2antll 460 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  ->  S  <Q  t )
5655adantr 261 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  S  <Q  t )
5756ad2antrr 457 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  S  <Q  t )
586ad5antr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  S  e.  Q. )
5939ad2antrr 457 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
t  e.  Q. )
601ad5antr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  F : N. --> Q. )
61 simplr 482 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  ->  m  e.  N. )
6260, 61ffvelrnd 5281 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( F `  m
)  e.  Q. )
63 nnnq 6492 . . . . . . . . . . . . . . . . 17  |-  ( m  e.  N.  ->  [ <. m ,  1o >. ]  ~Q  e.  Q. )
64 recclnq 6462 . . . . . . . . . . . . . . . . 17  |-  ( [
<. m ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
6561, 63, 643syl 17 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )
66 addclnq 6445 . . . . . . . . . . . . . . . 16  |-  ( ( ( F `  m
)  e.  Q.  /\  ( *Q `  [ <. m ,  1o >. ]  ~Q  )  e.  Q. )  ->  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
6762, 65, 66syl2anc 391 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )
68 ltanqg 6470 . . . . . . . . . . . . . . 15  |-  ( ( S  e.  Q.  /\  t  e.  Q.  /\  (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  e.  Q. )  ->  ( S  <Q  t  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t ) ) )
6958, 59, 67, 68syl3anc 1135 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( S  <Q  t  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t ) ) )
7057, 69mpbid 135 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  t ) )
71 simpr 103 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )
72 ltanqg 6470 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )  ->  (
z  <Q  w  <->  ( v  +Q  z )  <Q  (
v  +Q  w ) ) )
7372adantl 262 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  /\  ( z  e.  Q.  /\  w  e.  Q.  /\  v  e.  Q. )
)  ->  ( z  <Q  w  <->  ( v  +Q  z )  <Q  (
v  +Q  w ) ) )
7427ad2antrr 457 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
s  e.  Q. )
75 addcomnqg 6451 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  Q.  /\  w  e.  Q. )  ->  ( z  +Q  w
)  =  ( w  +Q  z ) )
7675adantl 262 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  /\  ( z  e.  Q.  /\  w  e.  Q. )
)  ->  ( z  +Q  w )  =  ( w  +Q  z ) )
7773, 67, 74, 59, 76caovord2d 5648 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t )  <Q 
( s  +Q  t
) ) )
7871, 77mpbid 135 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t )  <Q  (
s  +Q  t ) )
79 ltsonq 6468 . . . . . . . . . . . . . 14  |-  <Q  Or  Q.
8079, 34sotri 4698 . . . . . . . . . . . . 13  |-  ( ( ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  t )  /\  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  t )  <Q 
( s  +Q  t
) )  ->  (
( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  ( s  +Q  t
) )
8170, 78, 80syl2anc 391 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  (
s  +Q  t ) )
82 simpllr 486 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
r  =  ( s  +Q  t ) )
8381, 82breqtrrd 3787 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  /\  ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  <Q  s )  -> 
( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
8483ex 108 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  m  e.  N. )  ->  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s  ->  ( (
( F `  m
)  +Q  ( *Q
`  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
8584reximdva 2418 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( E. m  e. 
N.  ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  <Q 
s  ->  E. m  e.  N.  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q 
r ) )
8653, 85mpd 13 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. m  e.  N.  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
8750oveq1d 5505 . . . . . . . . . 10  |-  ( j  =  m  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S )  =  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S ) )
8887breq1d 3771 . . . . . . . . 9  |-  ( j  =  m  ->  (
( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r  <->  ( ( ( F `  m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
8988cbvrexv 2531 . . . . . . . 8  |-  ( E. j  e.  N.  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r  <->  E. m  e.  N.  ( ( ( F `
 m )  +Q  ( *Q `  [ <. m ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
9086, 89sylibr 137 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  r
)
91 breq2 3765 . . . . . . . . 9  |-  ( u  =  r  ->  (
( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u  <->  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
9291rexbidv 2324 . . . . . . . 8  |-  ( u  =  r  ->  ( E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u  <->  E. j  e.  N.  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  +Q  S ) 
<Q  r ) )
9392elrab 2695 . . . . . . 7  |-  ( r  e.  { u  e. 
Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u }  <->  ( r  e. 
Q.  /\  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q 
r ) )
9444, 90, 93sylanbrc 394 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
9594ex 108 . . . . 5  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( r  =  ( s  +Q  t )  ->  r  e.  {
u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } ) )
9695rexlimdvva 2437 . . . 4  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  ( E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t )  -> 
r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } ) )
9713, 96mpd 13 . . 3  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
9897ex 108 . 2  |-  ( ph  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  ->  r  e.  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } ) )
9998ssrdv 2948 1  |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  C_  { u  e.  Q.  |  E. j  e.  N.  ( ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  +Q  S )  <Q  u } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    /\ w3a 885    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307    C_ wss 2914   <.cop 3375   class class class wbr 3761   -->wf 4876   ` cfv 4880  (class class class)co 5490   2ndc2nd 5744   1oc1o 5972   [cec 6082   N.cnpi 6342    <N clti 6345    ~Q ceq 6349   Q.cnq 6350    +Q cplq 6352   *Qcrq 6354    <Q cltq 6355   P.cnp 6361    +P. cpp 6363
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4157  ax-setind 4247  ax-iinf 4289
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4090  df-on 4092  df-suc 4095  df-iom 4292  df-xp 4329  df-rel 4330  df-cnv 4331  df-co 4332  df-dm 4333  df-rn 4334  df-res 4335  df-ima 4336  df-iota 4845  df-fun 4882  df-fn 4883  df-f 4884  df-f1 4885  df-fo 4886  df-f1o 4887  df-fv 4888  df-ov 5493  df-oprab 5494  df-mpt2 5495  df-1st 5745  df-2nd 5746  df-recs 5898  df-irdg 5935  df-1o 5979  df-oadd 5983  df-omul 5984  df-er 6084  df-ec 6086  df-qs 6090  df-ni 6374  df-pli 6375  df-mi 6376  df-lti 6377  df-plpq 6414  df-mpq 6415  df-enq 6417  df-nqqs 6418  df-plqqs 6419  df-mqqs 6420  df-1nqqs 6421  df-rq 6422  df-ltnqqs 6423  df-inp 6536  df-iplp 6538
This theorem is referenced by:  caucvgprlemladdrl  6748
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