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Theorem caucvgprprlemlol 7448
Description: Lemma for caucvgprpr 7462. 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 7115 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4549 . . . 4  |-  ( s 
<Q  t  ->  ( s  e.  Q.  /\  t  e.  Q. ) )
32simpld 111 . . 3  |-  ( s 
<Q  t  ->  s  e. 
Q. )
433ad2ant2 984 . 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 7435 . . . . . 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 271 . . . . 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 985 . . . 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 1002 . . . . . . . . 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 7150 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
x  <Q  y  <->  ( z  +Q  x )  <Q  (
z  +Q  y ) ) )
1110adantl 273 . . . . . . . . . 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 477 . . . . . . . . . 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 113 . . . . . . . . . . . 12  |-  ( s 
<Q  t  ->  t  e. 
Q. )
14133ad2ant2 984 . . . . . . . . . . 11  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  t  e.  Q. )
1514ad2antrr 477 . . . . . . . . . 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 502 . . . . . . . . . . 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 7172 . . . . . . . . . . 11  |-  ( b  e.  N.  ->  [ <. b ,  1o >. ]  ~Q  e.  Q. )
18 recclnq 7142 . . . . . . . . . . 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 7131 . . . . . . . . . . 11  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  +Q  y
)  =  ( y  +Q  x ) )
2120adantl 273 . . . . . . . . . 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 5892 . . . . . . . . 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 146 . . . . . . . 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 7344 . . . . . . . 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 7346 . . . . . . . 8  |-  <P  Or  P.
27 ltrelpr 7255 . . . . . . . 8  |-  <P  C_  ( P.  X.  P. )
2826, 27sotri 4890 . . . . . . 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 414 . . . . . 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 114 . . . . 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 2506 . . . 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 3669 . . . . . . . . . . 11  |-  ( b  =  r  ->  <. b ,  1o >.  =  <. r ,  1o >. )
3433eceq1d 6417 . . . . . . . . . 10  |-  ( b  =  r  ->  [ <. b ,  1o >. ]  ~Q  =  [ <. r ,  1o >. ]  ~Q  )
3534fveq2d 5377 . . . . . . . . 9  |-  ( b  =  r  ->  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )
3635oveq2d 5742 . . . . . . . 8  |-  ( b  =  r  ->  (
s  +Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  )
)  =  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) )
3736breq2d 3905 . . . . . . 7  |-  ( b  =  r  ->  (
p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <->  p  <Q  ( s  +Q  ( *Q
`  [ <. r ,  1o >. ]  ~Q  )
) ) )
3837abbidv 2230 . . . . . 6  |-  ( b  =  r  ->  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) ) }  =  { p  |  p  <Q  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) } )
3936breq1d 3903 . . . . . . 7  |-  ( b  =  r  ->  (
( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  )
)  <Q  q  <->  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) )  <Q 
q ) )
4039abbidv 2230 . . . . . 6  |-  ( b  =  r  ->  { q  |  ( s  +Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) )  <Q 
q }  =  {
q  |  ( s  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } )
4138, 40opeq12d 3677 . . . . 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 5373 . . . . 5  |-  ( b  =  r  ->  ( F `  b )  =  ( F `  r ) )
4341, 42breq12d 3906 . . . 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 2627 . . 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 121 . 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 7435 . 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 411 1  |-  ( (
ph  /\  s  <Q  t  /\  t  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 943    = wceq 1312    e. wcel 1461   {cab 2099   A.wral 2388   E.wrex 2389   {crab 2392   <.cop 3494   class class class wbr 3893   -->wf 5075   ` cfv 5079  (class class class)co 5726   1stc1st 5988   1oc1o 6258   [cec 6379   N.cnpi 7022    <N clti 7025    ~Q ceq 7029   Q.cnq 7030    +Q cplq 7032   *Qcrq 7034    <Q cltq 7035   P.cnp 7041    +P. cpp 7043    <P cltp 7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-inp 7216  df-iltp 7220
This theorem is referenced by:  caucvgprprlemrnd  7451
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