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Theorem caucvgprlemcl 7452
Description: Lemma for caucvgpr 7458. 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 5389 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
54breq2d 3911 . . . . . 6  |-  ( j  =  a  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  a ) ) )
65cbvralv 2631 . . . . 5  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. a  e.  N.  A  <Q  ( F `  a ) )
73, 6sylib 121 . . . 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 3675 . . . . . . . . . . . . 13  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
109eceq1d 6433 . . . . . . . . . . . 12  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1110fveq2d 5393 . . . . . . . . . . 11  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1211oveq2d 5758 . . . . . . . . . 10  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1312, 4breq12d 3912 . . . . . . . . 9  |-  ( j  =  a  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
1413cbvrexv 2632 . . . . . . . 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 2645 . . . . . 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 5760 . . . . . . . . . 10  |-  ( j  =  a  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1817breq1d 3909 . . . . . . . . 9  |-  ( j  =  a  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u ) )
1918cbvrexv 2632 . . . . . . . 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 2645 . . . . . 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 3680 . . . . 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 2138 . . . 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 7444 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
25 ssrab2 3152 . . . . . 6  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  C_  Q.
26 nqex 7139 . . . . . . 7  |-  Q.  e.  _V
2726elpw2 4052 . . . . . 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 145 . . . . 5  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.
29 ssrab2 3152 . . . . . 6  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  C_  Q.
3026elpw2 4052 . . . . . 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 145 . . . . 5  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  ~P Q.
32 opelxpi 4541 . . . . 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 422 . . . 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 2190 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
3524, 34jctil 310 . 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 7449 . . 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 3902 . . . . . . 7  |-  ( n  =  c  ->  (
n  <N  k  <->  c  <N  k ) )
38 fveq2 5389 . . . . . . . . 9  |-  ( n  =  c  ->  ( F `  n )  =  ( F `  c ) )
39 opeq1 3675 . . . . . . . . . . . 12  |-  ( n  =  c  ->  <. n ,  1o >.  =  <. c ,  1o >. )
4039eceq1d 6433 . . . . . . . . . . 11  |-  ( n  =  c  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
4140fveq2d 5393 . . . . . . . . . 10  |-  ( n  =  c  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
4241oveq2d 5758 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4338, 42breq12d 3912 . . . . . . . 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 5760 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4544breq2d 3911 . . . . . . . 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 464 . . . . . . 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 233 . . . . . 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 3903 . . . . . . 7  |-  ( k  =  d  ->  (
c  <N  k  <->  c  <N  d ) )
49 fveq2 5389 . . . . . . . . . 10  |-  ( k  =  d  ->  ( F `  k )  =  ( F `  d ) )
5049oveq1d 5757 . . . . . . . . 9  |-  ( k  =  d  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  =  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
5150breq2d 3911 . . . . . . . 8  |-  ( k  =  d  ->  (
( F `  c
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <->  ( F `  c )  <Q  (
( F `  d
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
5249breq1d 3909 . . . . . . . 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 464 . . . . . . 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 233 . . . . . 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 2639 . . . . 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 121 . . . 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 7450 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
581, 2, 7, 23caucvgprlemloc 7451 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
5936, 57, 583jca 1146 . 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 7252 . 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 413 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 682    /\ w3a 947    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041   ~Pcpw 3480   <.cop 3500   class class class wbr 3899    X. cxp 4507   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061   P.cnp 7067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242
This theorem is referenced by:  caucvgprlemladdfu  7453  caucvgprlemladdrl  7454  caucvgprlem1  7455  caucvgprlem2  7456  caucvgpr  7458
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