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Theorem caucvgprprlemlol 6950
Description: Lemma for caucvgprpr 6964. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 21-Dec-2020.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
Assertion
Ref Expression
caucvgprprlemlol  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Distinct variable groups:    A, m    m, F    F, l    u, F, r    p, l, s   
q, l, s, r   
t, l, p    u, q, s, r    u, p, t, r    ph, r    r, q, t
Allowed substitution hints:    ph( u, t, k, m, n, s, q, p, l)    A( u, t, k, n, s, r, q, p, l)    F( t, k, n, s, q, p)    L( u, t, k, m, n, s, r, q, p, l)

Proof of Theorem caucvgprprlemlol
Dummy variables  b  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6617 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4418 . . . 4  |-  ( s 
<Q  t  ->  ( s  e.  Q.  /\  t  e.  Q. ) )
32simpld 110 . . 3  |-  ( s 
<Q  t  ->  s  e. 
Q. )
433ad2ant2 961 . 2  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  Q. )
5 caucvgprpr.lim . . . . . . 7  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
65caucvgprprlemell 6937 . . . . . 6  |-  ( t  e.  ( 1st `  L
)  <->  ( t  e. 
Q.  /\  E. b  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) ) )
76simprbi 269 . . . . 5  |-  ( t  e.  ( 1st `  L
)  ->  E. b  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )
873ad2ant3 962 . . . 4  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  E. b  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )
9 simpll2 979 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  s  <Q  t )
10 ltanqg 6652 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  +Q  x )  <Q  (
z  +Q  y ) ) )
1110adantl 271 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )  /\  (
x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. ) )  -> 
( x  <Q  y  <->  ( z  +Q  x ) 
<Q  ( z  +Q  y
) ) )
124ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  s  e.  Q. )
132simprd 112 . . . . . . . . . . . 12  |-  ( s 
<Q  t  ->  t  e. 
Q. )
14133ad2ant2 961 . . . . . . . . . . 11  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  t  e.  Q. )
1514ad2antrr 472 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  t  e.  Q. )
16 simplr 497 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  b  e.  N. )
17 nnnq 6674 . . . . . . . . . . 11  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 6644 . . . . . . . . . . 11  |-  ( [
<. b ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e.  Q. )
1916, 17, 183syl 17 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  e. 
Q. )
20 addcomnqg 6633 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  =  ( y  +Q  x ) )
2120adantl 271 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )  /\  (
x  e.  Q.  /\  y  e.  Q. )
)  ->  ( x  +Q  y )  =  ( y  +Q  x ) )
2211, 12, 15, 19, 21caovord2d 5701 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  ( s  <Q  t  <->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) ) )
239, 22mpbid 145 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) )
24 ltnqpri 6846 . . . . . . . 8  |-  ( ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  ->  <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >. )
2523, 24syl 14 . . . . . . 7  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >. )
26 ltsopr 6848 . . . . . . . 8  |-  <P  Or  P.
27 ltrelpr 6757 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
2826, 27sotri 4750 . . . . . . 7  |-  ( (
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )
)
2925, 28sylancom 411 . . . . . 6  |-  ( ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  /\  <. { p  |  p  <Q  ( t  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
)  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )
)
3029ex 113 . . . . 5  |-  ( ( ( ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L
) )  /\  b  e.  N. )  ->  ( <. { p  |  p 
<Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )  -> 
<. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )
3130reximdva 2464 . . . 4  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  ( E. b  e.  N.  <. { p  |  p  <Q  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( t  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  <P  ( F `  b )  ->  E. b  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )
) )
328, 31mpd 13 . . 3  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  E. b  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  b
) )
33 opeq1 3578 . . . . . . . . . . 11  |-  ( b  =  r  ->  <. b ,  1o >.  =  <. r ,  1o >. )
3433eceq1d 6208 . . . . . . . . . 10  |-  ( b  =  r  ->  [ <. b ,  1o >. ]  ~Q  =  [ <. r ,  1o >. ]  ~Q  )
3534fveq2d 5213 . . . . . . . . 9  |-  ( b  =  r  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )
3635oveq2d 5559 . . . . . . . 8  |-  ( b  =  r  ->  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
3736breq2d 3805 . . . . . . 7  |-  ( b  =  r  ->  (
p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
3837abbidv 2197 . . . . . 6  |-  ( b  =  r  ->  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
3936breq1d 3803 . . . . . . 7  |-  ( b  =  r  ->  (
( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
4039abbidv 2197 . . . . . 6  |-  ( b  =  r  ->  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
4138, 40opeq12d 3586 . . . . 5  |-  ( b  =  r  ->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q } >.  =  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >. )
42 fveq2 5209 . . . . 5  |-  ( b  =  r  ->  ( F `  b )  =  ( F `  r ) )
4341, 42breq12d 3806 . . . 4  |-  ( b  =  r  ->  ( <. { p  |  p 
<Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )  <->  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
) )
4443cbvrexv 2579 . . 3  |-  ( E. b  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  b )  <->  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r )
)
4532, 44sylib 120 . 2  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) )
465caucvgprprlemell 6937 . 2  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. r  e.  N.  <. { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } ,  { q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
)  <Q  q } >.  <P 
( F `  r
) ) )
474, 45, 46sylanbrc 408 1  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3409   class class class wbr 3793   -->wf 4928   ` cfv 4932  (class class class)co 5543   1stc1st 5796   1oc1o 6058   [cec 6170   N.cnpi 6524    <N clti 6527    ~Q ceq 6531   Q.cnq 6532    +Q cplq 6534   *Qcrq 6536    <Q cltq 6537   P.cnp 6543    +P. cpp 6545    <P cltp 6547
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-inp 6718  df-iltp 6722
This theorem is referenced by:  caucvgprprlemrnd  6953
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