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Theorem caucvgprlemcl 6832
Description: Lemma for caucvgpr 6838. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-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 } >.
Assertion
Ref Expression
caucvgprlemcl  |-  ( ph  ->  L  e.  P. )
Distinct variable groups:    A, j    j, F, l    u, F, j   
n, F, k    j,
k, L    k, n
Allowed substitution hints:    ph( u, j, k, n, l)    A( u, k, n, l)    L( u, n, l)

Proof of Theorem caucvgprlemcl
Dummy variables  s  a  c  d  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . 4  |-  ( ph  ->  F : N. --> Q. )
2 caucvgpr.cau . . . 4  |-  ( 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 . . . . 5  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
4 fveq2 5206 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
54breq2d 3804 . . . . . 6  |-  ( j  =  a  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  a ) ) )
65cbvralv 2550 . . . . 5  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. a  e.  N.  A  <Q  ( F `  a ) )
73, 6sylib 131 . . . 4  |-  ( ph  ->  A. a  e.  N.  A  <Q  ( F `  a ) )
8 caucvgpr.lim . . . . 5  |-  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 } >.
9 opeq1 3577 . . . . . . . . . . . . 13  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
109eceq1d 6173 . . . . . . . . . . . 12  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1110fveq2d 5210 . . . . . . . . . . 11  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1211oveq2d 5556 . . . . . . . . . 10  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1312, 4breq12d 3805 . . . . . . . . 9  |-  ( j  =  a  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
1413cbvrexv 2551 . . . . . . . 8  |-  ( E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) )
1514a1i 9 . . . . . . 7  |-  ( l  e.  Q.  ->  ( E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
1615rabbiia 2564 . . . . . 6  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  =  { l  e.  Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( F `  a ) }
174, 11oveq12d 5558 . . . . . . . . . 10  |-  ( j  =  a  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1817breq1d 3802 . . . . . . . . 9  |-  ( j  =  a  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u ) )
1918cbvrexv 2551 . . . . . . . 8  |-  ( E. j  e.  N.  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. a  e.  N.  ( ( F `
 a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u )
2019a1i 9 . . . . . . 7  |-  ( u  e.  Q.  ->  ( E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. a  e.  N.  ( ( F `
 a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u ) )
2120rabbiia 2564 . . . . . 6  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  =  {
u  e.  Q.  |  E. a  e.  N.  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u }
2216, 21opeq12i 3582 . . . . 5  |-  <. { 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 } >.  = 
<. { l  e.  Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( F `  a ) } ,  { u  e.  Q.  |  E. a  e.  N.  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >.
238, 22eqtri 2076 . . . 4  |-  L  = 
<. { l  e.  Q.  |  E. a  e.  N.  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  ( F `  a ) } ,  { u  e.  Q.  |  E. a  e.  N.  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  )
)  <Q  u } >.
241, 2, 7, 23caucvgprlemm 6824 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
25 ssrab2 3053 . . . . . 6  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  C_  Q.
26 nqex 6519 . . . . . . 7  |-  Q.  e.  _V
2726elpw2 3939 . . . . . 6  |-  ( { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.  <->  { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  C_  Q. )
2825, 27mpbir 138 . . . . 5  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.
29 ssrab2 3053 . . . . . 6  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  C_  Q.
3026elpw2 3939 . . . . . 6  |-  ( { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }  e.  ~P Q.  <->  { u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }  C_  Q. )
3129, 30mpbir 138 . . . . 5  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  ~P Q.
32 opelxpi 4404 . . . . 5  |-  ( ( { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.  /\  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  ~P Q. )  ->  <. { 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 } >.  e.  ( ~P Q.  X.  ~P Q. ) )
3328, 31, 32mp2an 410 . . . 4  |-  <. { 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 } >.  e.  ( ~P Q.  X.  ~P Q. )
348, 33eqeltri 2126 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
3524, 34jctil 299 . 2  |-  ( ph  ->  ( L  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. s  e.  Q.  s  e.  ( 1st `  L )  /\  E. r  e.  Q.  r  e.  ( 2nd `  L
) ) ) )
361, 2, 7, 23caucvgprlemrnd 6829 . . 3  |-  ( ph  ->  ( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) ) )
37 breq1 3795 . . . . . . 7  |-  ( n  =  c  ->  (
n  <N  k  <->  c  <N  k ) )
38 fveq2 5206 . . . . . . . . 9  |-  ( n  =  c  ->  ( F `  n )  =  ( F `  c ) )
39 opeq1 3577 . . . . . . . . . . . 12  |-  ( n  =  c  ->  <. n ,  1o >.  =  <. c ,  1o >. )
4039eceq1d 6173 . . . . . . . . . . 11  |-  ( n  =  c  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
4140fveq2d 5210 . . . . . . . . . 10  |-  ( n  =  c  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
4241oveq2d 5556 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4338, 42breq12d 3805 . . . . . . . 8  |-  ( n  =  c  ->  (
( F `  n
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  c )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
4438, 41oveq12d 5558 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4544breq2d 3804 . . . . . . . 8  |-  ( n  =  c  ->  (
( F `  k
)  <Q  ( ( F `
 n )  +Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) )  <->  ( F `  k )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
4643, 45anbi12d 450 . . . . . . 7  |-  ( n  =  c  ->  (
( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  c )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) ) )
4737, 46imbi12d 227 . . . . . 6  |-  ( n  =  c  ->  (
( n  <N  k  ->  ( ( F `  n )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
) ) )  <->  ( c  <N  k  ->  ( ( F `  c )  <Q  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) ) ) )
48 breq2 3796 . . . . . . 7  |-  ( k  =  d  ->  (
c  <N  k  <->  c  <N  d ) )
49 fveq2 5206 . . . . . . . . . 10  |-  ( k  =  d  ->  ( F `  k )  =  ( F `  d ) )
5049oveq1d 5555 . . . . . . . . 9  |-  ( k  =  d  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  =  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
5150breq2d 3804 . . . . . . . 8  |-  ( k  =  d  ->  (
( F `  c
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <->  ( F `  c )  <Q  (
( F `  d
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
5249breq1d 3802 . . . . . . . 8  |-  ( k  =  d  ->  (
( F `  k
)  <Q  ( ( F `
 c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <->  ( F `  d )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
5351, 52anbi12d 450 . . . . . . 7  |-  ( k  =  d  ->  (
( ( F `  c )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) )  <->  ( ( F `  c )  <Q  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  d )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) ) )
5448, 53imbi12d 227 . . . . . 6  |-  ( k  =  d  ->  (
( c  <N  k  ->  ( ( F `  c )  <Q  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  k )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )  <->  ( c  <N  d  ->  ( ( F `  c )  <Q  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  d )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) ) ) )
5547, 54cbvral2v 2558 . . . . 5  |-  ( 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  )
) ) )  <->  A. c  e.  N.  A. d  e. 
N.  ( c  <N 
d  ->  ( ( F `  c )  <Q  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
)  /\  ( F `  d )  <Q  (
( F `  c
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) ) )
562, 55sylib 131 . . . 4  |-  ( ph  ->  A. c  e.  N.  A. d  e.  N.  (
c  <N  d  ->  (
( F `  c
)  <Q  ( ( F `
 d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  /\  ( F `  d ) 
<Q  ( ( F `  c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  )
) ) ) )
571, 56, 7, 23caucvgprlemdisj 6830 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
581, 2, 7, 23caucvgprlemloc 6831 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
5936, 57, 583jca 1095 . 2  |-  ( ph  ->  ( ( A. s  e.  Q.  ( s  e.  ( 1st `  L
)  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) )  /\  A. s  e.  Q.  -.  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  /\  A. s  e.  Q.  A. r  e.  Q.  ( s  <Q 
r  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) ) ) )
60 elnp1st2nd 6632 . 2  |-  ( L  e.  P.  <->  ( ( L  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )  /\  (
( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) )  /\  A. s  e.  Q.  -.  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  /\  A. s  e.  Q.  A. r  e.  Q.  ( s  <Q 
r  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) ) ) ) )
6135, 59, 60sylanbrc 402 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324   {crab 2327    C_ wss 2945   ~Pcpw 3387   <.cop 3406   class class class wbr 3792    X. cxp 4371   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   1oc1o 6025   [cec 6135   N.cnpi 6428    <N clti 6431    ~Q ceq 6435   Q.cnq 6436    +Q cplq 6438   *Qcrq 6440    <Q cltq 6441   P.cnp 6447
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-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-inp 6622
This theorem is referenced by:  caucvgprlemladdfu  6833  caucvgprlemladdrl  6834  caucvgprlem1  6835  caucvgprlem2  6836  caucvgpr  6838
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