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Theorem caucvgprlemcl 6746
Description: Lemma for caucvgpr 6752. 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 5156 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
54breq2d 3773 . . . . . 6  |-  ( j  =  a  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  a ) ) )
65cbvralv 2530 . . . . 5  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. a  e.  N.  A  <Q  ( F `  a ) )
73, 6sylib 127 . . . 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 3546 . . . . . . . . . . . . 13  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
109eceq1d 6120 . . . . . . . . . . . 12  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1110fveq2d 5160 . . . . . . . . . . 11  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1211oveq2d 5506 . . . . . . . . . 10  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1312, 4breq12d 3774 . . . . . . . . 9  |-  ( j  =  a  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
1413cbvrexv 2531 . . . . . . . 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 2544 . . . . . 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 5508 . . . . . . . . . 10  |-  ( j  =  a  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1817breq1d 3771 . . . . . . . . 9  |-  ( j  =  a  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u ) )
1918cbvrexv 2531 . . . . . . . 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 2544 . . . . . 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 3551 . . . . 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 2060 . . . 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 6738 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
25 ssrab2 3022 . . . . . 6  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  C_  Q.
26 nqex 6433 . . . . . . 7  |-  Q.  e.  _V
2726elpw2 3908 . . . . . 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 134 . . . . 5  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.
29 ssrab2 3022 . . . . . 6  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  C_  Q.
3026elpw2 3908 . . . . . 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 134 . . . . 5  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  ~P Q.
32 opelxpi 4354 . . . . 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 402 . . . 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 2110 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
3524, 34jctil 295 . 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 6743 . . 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 3764 . . . . . . 7  |-  ( n  =  c  ->  (
n  <N  k  <->  c  <N  k ) )
38 fveq2 5156 . . . . . . . . 9  |-  ( n  =  c  ->  ( F `  n )  =  ( F `  c ) )
39 opeq1 3546 . . . . . . . . . . . 12  |-  ( n  =  c  ->  <. n ,  1o >.  =  <. c ,  1o >. )
4039eceq1d 6120 . . . . . . . . . . 11  |-  ( n  =  c  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
4140fveq2d 5160 . . . . . . . . . 10  |-  ( n  =  c  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
4241oveq2d 5506 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4338, 42breq12d 3774 . . . . . . . 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 5508 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4544breq2d 3773 . . . . . . . 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 442 . . . . . . 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 223 . . . . . 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 3765 . . . . . . 7  |-  ( k  =  d  ->  (
c  <N  k  <->  c  <N  d ) )
49 fveq2 5156 . . . . . . . . . 10  |-  ( k  =  d  ->  ( F `  k )  =  ( F `  d ) )
5049oveq1d 5505 . . . . . . . . 9  |-  ( k  =  d  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  =  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
5150breq2d 3773 . . . . . . . 8  |-  ( k  =  d  ->  (
( F `  c
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <->  ( F `  c )  <Q  (
( F `  d
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
5249breq1d 3771 . . . . . . . 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 442 . . . . . . 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 223 . . . . . 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 2538 . . . . 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 127 . . . 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 6744 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
581, 2, 7, 23caucvgprlemloc 6745 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
5936, 57, 583jca 1084 . 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 6546 . 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 394 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 97    <-> wb 98    \/ wo 629    /\ w3a 885    = wceq 1243    e. wcel 1393   A.wral 2303   E.wrex 2304   {crab 2307    C_ wss 2914   ~Pcpw 3356   <.cop 3375   class class class wbr 3761    X. cxp 4321   -->wf 4876   ` cfv 4880  (class class class)co 5490   1stc1st 5743   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
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
This theorem is referenced by:  caucvgprlemladdfu  6747  caucvgprlemladdrl  6748  caucvgprlem1  6749  caucvgprlem2  6750  caucvgpr  6752
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