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Theorem caucvgprlemcl 7650
Description: Lemma for caucvgpr 7656. 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 5507 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
54breq2d 4010 . . . . . 6  |-  ( j  =  a  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  a ) ) )
65cbvralv 2701 . . . . 5  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. a  e.  N.  A  <Q  ( F `  a ) )
73, 6sylib 122 . . . 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 3774 . . . . . . . . . . . . 13  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
109eceq1d 6561 . . . . . . . . . . . 12  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1110fveq2d 5511 . . . . . . . . . . 11  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1211oveq2d 5881 . . . . . . . . . 10  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1312, 4breq12d 4011 . . . . . . . . 9  |-  ( j  =  a  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
1413cbvrexv 2702 . . . . . . . 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 2720 . . . . . 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 5883 . . . . . . . . . 10  |-  ( j  =  a  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1817breq1d 4008 . . . . . . . . 9  |-  ( j  =  a  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u ) )
1918cbvrexv 2702 . . . . . . . 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 2720 . . . . . 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 3779 . . . . 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 2196 . . . 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 7642 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
25 ssrab2 3238 . . . . . 6  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  C_  Q.
26 nqex 7337 . . . . . . 7  |-  Q.  e.  _V
2726elpw2 4152 . . . . . 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 146 . . . . 5  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.
29 ssrab2 3238 . . . . . 6  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  C_  Q.
3026elpw2 4152 . . . . . 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 146 . . . . 5  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  ~P Q.
32 opelxpi 4652 . . . . 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 426 . . . 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 2248 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
3524, 34jctil 312 . 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 7647 . . 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 4001 . . . . . . 7  |-  ( n  =  c  ->  (
n  <N  k  <->  c  <N  k ) )
38 fveq2 5507 . . . . . . . . 9  |-  ( n  =  c  ->  ( F `  n )  =  ( F `  c ) )
39 opeq1 3774 . . . . . . . . . . . 12  |-  ( n  =  c  ->  <. n ,  1o >.  =  <. c ,  1o >. )
4039eceq1d 6561 . . . . . . . . . . 11  |-  ( n  =  c  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
4140fveq2d 5511 . . . . . . . . . 10  |-  ( n  =  c  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
4241oveq2d 5881 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4338, 42breq12d 4011 . . . . . . . 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 5883 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4544breq2d 4010 . . . . . . . 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 473 . . . . . . 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 234 . . . . . 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 4002 . . . . . . 7  |-  ( k  =  d  ->  (
c  <N  k  <->  c  <N  d ) )
49 fveq2 5507 . . . . . . . . . 10  |-  ( k  =  d  ->  ( F `  k )  =  ( F `  d ) )
5049oveq1d 5880 . . . . . . . . 9  |-  ( k  =  d  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  =  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
5150breq2d 4010 . . . . . . . 8  |-  ( k  =  d  ->  (
( F `  c
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <->  ( F `  c )  <Q  (
( F `  d
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
5249breq1d 4008 . . . . . . . 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 473 . . . . . . 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 234 . . . . . 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 2714 . . . . 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 122 . . . 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 7648 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
581, 2, 7, 23caucvgprlemloc 7649 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
5936, 57, 583jca 1177 . 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 7450 . 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 417 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    /\ w3a 978    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454   {crab 2457    C_ wss 3127   ~Pcpw 3572   <.cop 3592   class class class wbr 3998    X. cxp 4618   -->wf 5204   ` cfv 5208  (class class class)co 5865   1stc1st 6129   2ndc2nd 6130   1oc1o 6400   [cec 6523   N.cnpi 7246    <N clti 7249    ~Q ceq 7253   Q.cnq 7254    +Q cplq 7256   *Qcrq 7258    <Q cltq 7259   P.cnp 7265
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-inp 7440
This theorem is referenced by:  caucvgprlemladdfu  7651  caucvgprlemladdrl  7652  caucvgprlem1  7653  caucvgprlem2  7654  caucvgpr  7656
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