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Theorem caucvgprlemcl 7233
Description: Lemma for caucvgpr 7239. 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 5305 . . . . . . 7  |-  ( j  =  a  ->  ( F `  j )  =  ( F `  a ) )
54breq2d 3857 . . . . . 6  |-  ( j  =  a  ->  ( A  <Q  ( F `  j )  <->  A  <Q  ( F `  a ) ) )
65cbvralv 2590 . . . . 5  |-  ( A. j  e.  N.  A  <Q  ( F `  j
)  <->  A. a  e.  N.  A  <Q  ( F `  a ) )
73, 6sylib 120 . . . 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 3622 . . . . . . . . . . . . 13  |-  ( j  =  a  ->  <. j ,  1o >.  =  <. a ,  1o >. )
109eceq1d 6326 . . . . . . . . . . . 12  |-  ( j  =  a  ->  [ <. j ,  1o >. ]  ~Q  =  [ <. a ,  1o >. ]  ~Q  )
1110fveq2d 5309 . . . . . . . . . . 11  |-  ( j  =  a  ->  ( *Q `  [ <. j ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )
1211oveq2d 5668 . . . . . . . . . 10  |-  ( j  =  a  ->  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1312, 4breq12d 3858 . . . . . . . . 9  |-  ( j  =  a  ->  (
( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j )  <->  ( l  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q 
( F `  a
) ) )
1413cbvrexv 2591 . . . . . . . 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 2604 . . . . . 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 5670 . . . . . . . . . 10  |-  ( j  =  a  ->  (
( F `  j
)  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  =  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) ) )
1817breq1d 3855 . . . . . . . . 9  |-  ( j  =  a  ->  (
( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  a )  +Q  ( *Q `  [ <. a ,  1o >. ]  ~Q  ) )  <Q  u ) )
1918cbvrexv 2591 . . . . . . . 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 2604 . . . . . 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 3627 . . . . 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 2108 . . . 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 7225 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
25 ssrab2 3106 . . . . . 6  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  C_  Q.
26 nqex 6920 . . . . . . 7  |-  Q.  e.  _V
2726elpw2 3993 . . . . . 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 144 . . . . 5  |-  { l  e.  Q.  |  E. j  e.  N.  (
l  +Q  ( *Q
`  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }  e.  ~P Q.
29 ssrab2 3106 . . . . . 6  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  C_  Q.
3026elpw2 3993 . . . . . 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 144 . . . . 5  |-  { u  e.  Q.  |  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u }  e.  ~P Q.
32 opelxpi 4469 . . . . 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 417 . . . 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 2160 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
3524, 34jctil 305 . 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 7230 . . 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 3848 . . . . . . 7  |-  ( n  =  c  ->  (
n  <N  k  <->  c  <N  k ) )
38 fveq2 5305 . . . . . . . . 9  |-  ( n  =  c  ->  ( F `  n )  =  ( F `  c ) )
39 opeq1 3622 . . . . . . . . . . . 12  |-  ( n  =  c  ->  <. n ,  1o >.  =  <. c ,  1o >. )
4039eceq1d 6326 . . . . . . . . . . 11  |-  ( n  =  c  ->  [ <. n ,  1o >. ]  ~Q  =  [ <. c ,  1o >. ]  ~Q  )
4140fveq2d 5309 . . . . . . . . . 10  |-  ( n  =  c  ->  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )
4241oveq2d 5668 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4338, 42breq12d 3858 . . . . . . . 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 5670 . . . . . . . . 9  |-  ( n  =  c  ->  (
( F `  n
)  +Q  ( *Q
`  [ <. n ,  1o >. ]  ~Q  )
)  =  ( ( F `  c )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
4544breq2d 3857 . . . . . . . 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 457 . . . . . . 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 232 . . . . . 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 3849 . . . . . . 7  |-  ( k  =  d  ->  (
c  <N  k  <->  c  <N  d ) )
49 fveq2 5305 . . . . . . . . . 10  |-  ( k  =  d  ->  ( F `  k )  =  ( F `  d ) )
5049oveq1d 5667 . . . . . . . . 9  |-  ( k  =  d  ->  (
( F `  k
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
)  =  ( ( F `  d )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) ) )
5150breq2d 3857 . . . . . . . 8  |-  ( k  =  d  ->  (
( F `  c
)  <Q  ( ( F `
 k )  +Q  ( *Q `  [ <. c ,  1o >. ]  ~Q  ) )  <->  ( F `  c )  <Q  (
( F `  d
)  +Q  ( *Q
`  [ <. c ,  1o >. ]  ~Q  )
) ) )
5249breq1d 3855 . . . . . . . 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 457 . . . . . . 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 232 . . . . . 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 2598 . . . . 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 120 . . . 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 7231 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
581, 2, 7, 23caucvgprlemloc 7232 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
5936, 57, 583jca 1123 . 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 7033 . 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 408 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2999   ~Pcpw 3429   <.cop 3449   class class class wbr 3845    X. cxp 4436   -->wf 5011   ` cfv 5015  (class class class)co 5652   1stc1st 5909   2ndc2nd 5910   1oc1o 6174   [cec 6288   N.cnpi 6829    <N clti 6832    ~Q ceq 6836   Q.cnq 6837    +Q cplq 6839   *Qcrq 6841    <Q cltq 6842   P.cnp 6848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-1o 6181  df-oadd 6185  df-omul 6186  df-er 6290  df-ec 6292  df-qs 6296  df-ni 6861  df-pli 6862  df-mi 6863  df-lti 6864  df-plpq 6901  df-mpq 6902  df-enq 6904  df-nqqs 6905  df-plqqs 6906  df-mqqs 6907  df-1nqqs 6908  df-rq 6909  df-ltnqqs 6910  df-inp 7023
This theorem is referenced by:  caucvgprlemladdfu  7234  caucvgprlemladdrl  7235  caucvgprlem1  7236  caucvgprlem2  7237  caucvgpr  7239
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