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Theorem caucvgpr 6761
Description: A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, the HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within  1  /  n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis). We also specify that every term needs to be larger than a fraction  A, to avoid the case where we have positive terms which "converge" to zero (which is not a positive real). (Contributed by Jim Kingdon, 18-Jun-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 ) )
Assertion
Ref Expression
caucvgpr  |-  ( ph  ->  E. y  e.  P.  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
Distinct variable groups:    A, j    j, F, k, n, l, u, x, y    ph, j,
k, x
Allowed substitution hints:    ph( y, u, n, l)    A( x, y, u, k, n, l)

Proof of Theorem caucvgpr
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . 3  |-  ( ph  ->  F : N. --> Q. )
2 caucvgpr.cau . . 3  |-  ( 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 . . 3  |-  ( ph  ->  A. j  e.  N.  A  <Q  ( F `  j ) )
4 opeq1 3546 . . . . . . . . . . 11  |-  ( z  =  j  ->  <. z ,  1o >.  =  <. j ,  1o >. )
54eceq1d 6129 . . . . . . . . . 10  |-  ( z  =  j  ->  [ <. z ,  1o >. ]  ~Q  =  [ <. j ,  1o >. ]  ~Q  )
65fveq2d 5169 . . . . . . . . 9  |-  ( z  =  j  ->  ( *Q `  [ <. z ,  1o >. ]  ~Q  )  =  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )
76oveq2d 5515 . . . . . . . 8  |-  ( z  =  j  ->  (
l  +Q  ( *Q
`  [ <. z ,  1o >. ]  ~Q  )
)  =  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
8 fveq2 5165 . . . . . . . 8  |-  ( z  =  j  ->  ( F `  z )  =  ( F `  j ) )
97, 8breq12d 3774 . . . . . . 7  |-  ( z  =  j  ->  (
( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z )  <->  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
109cbvrexv 2531 . . . . . 6  |-  ( E. z  e.  N.  (
l  +Q  ( *Q
`  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z )  <->  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) )
1110a1i 9 . . . . 5  |-  ( l  e.  Q.  ->  ( E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z )  <->  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q 
( F `  j
) ) )
1211rabbiia 2544 . . . 4  |-  { l  e.  Q.  |  E. z  e.  N.  (
l  +Q  ( *Q
`  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) }  =  { l  e.  Q.  |  E. j  e.  N.  ( l  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  ( F `  j ) }
138, 6oveq12d 5517 . . . . . . . 8  |-  ( z  =  j  ->  (
( F `  z
)  +Q  ( *Q
`  [ <. z ,  1o >. ]  ~Q  )
)  =  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) ) )
1413breq1d 3771 . . . . . . 7  |-  ( z  =  j  ->  (
( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u  <->  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u ) )
1514cbvrexv 2531 . . . . . 6  |-  ( E. z  e.  N.  (
( F `  z
)  +Q  ( *Q
`  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u )
1615a1i 9 . . . . 5  |-  ( u  e.  Q.  ->  ( E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u  <->  E. j  e.  N.  ( ( F `
 j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  ) )  <Q  u ) )
1716rabbiia 2544 . . . 4  |-  { u  e.  Q.  |  E. z  e.  N.  ( ( F `
 z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q  u }  =  {
u  e.  Q.  |  E. j  e.  N.  ( ( F `  j )  +Q  ( *Q `  [ <. j ,  1o >. ]  ~Q  )
)  <Q  u }
1812, 17opeq12i 3551 . . 3  |-  <. { l  e.  Q.  |  E. z  e.  N.  (
l  +Q  ( *Q
`  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  = 
<. { 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 } >.
191, 2, 3, 18caucvgprlemcl 6755 . 2  |-  ( ph  -> 
<. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  e. 
P. )
201, 2, 3, 18caucvgprlemlim 6760 . 2  |-  ( ph  ->  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  ( <. { l  |  l 
<Q  ( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) ) )
21 oveq1 5506 . . . . . . . 8  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x  <Q  u } >. )  =  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. ) )
2221breq2d 3773 . . . . . . 7  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  (
y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x  <Q  u } >. )  <->  <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. ) ) )
23 breq1 3764 . . . . . . 7  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( y  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. 
<-> 
<. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) )
2422, 23anbi12d 442 . . . . . 6  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( ( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( y  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  y  <P 
<. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. )  <->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) ) )
2524imbi2d 219 . . . . 5  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) )  <->  ( j  <N  k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) ) ) )
2625rexralbidv 2347 . . . 4  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  (
y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x  <Q  u } >. )  /\  y  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) )  <->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) ) ) )
2726ralbidv 2323 . . 3  |-  ( y  =  <. { l  e. 
Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  ) )  <Q 
( F `  z
) } ,  {
u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  -> 
( A. x  e. 
Q.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  ( <. { l  |  l 
<Q  ( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) )  <->  A. x  e.  Q.  E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) ) ) )
2827rspcev 2653 . 2  |-  ( (
<. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  e. 
P.  /\  A. x  e.  Q.  E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( <. { l  |  l  <Q  ( F `  k ) } ,  { u  |  ( F `  k )  <Q  u } >.  <P  ( <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  +P. 
<. { l  |  l 
<Q  x } ,  {
u  |  x  <Q  u } >. )  /\  <. { l  e.  Q.  |  E. z  e.  N.  ( l  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  ( F `  z ) } ,  { u  e.  Q.  |  E. z  e.  N.  ( ( F `  z )  +Q  ( *Q `  [ <. z ,  1o >. ]  ~Q  )
)  <Q  u } >.  <P  <. { l  |  l 
<Q  ( ( F `  k )  +Q  x
) } ,  {
u  |  ( ( F `  k )  +Q  x )  <Q  u } >. ) ) )  ->  E. y  e.  P.  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
2919, 20, 28syl2anc 391 1  |-  ( ph  ->  E. y  e.  P.  A. x  e.  Q.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( <. { l  |  l  <Q 
( F `  k
) } ,  {
u  |  ( F `
 k )  <Q  u } >.  <P  ( y  +P.  <. { l  |  l  <Q  x } ,  { u  |  x 
<Q  u } >. )  /\  y  <P  <. { l  |  l  <Q  (
( F `  k
)  +Q  x ) } ,  { u  |  ( ( F `
 k )  +Q  x )  <Q  u } >. ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307   <.cop 3375   class class class wbr 3761   -->wf 4885   ` cfv 4889  (class class class)co 5499   1oc1o 5981   [cec 6091   N.cnpi 6351    <N clti 6354    ~Q ceq 6358   Q.cnq 6359    +Q cplq 6361   *Qcrq 6363    <Q cltq 6364   P.cnp 6370    +P. cpp 6372    <P cltp 6374
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 4166  ax-setind 4256  ax-iinf 4298
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 4099  df-on 4101  df-suc 4104  df-iom 4301  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-rn 4343  df-res 4344  df-ima 4345  df-iota 4854  df-fun 4891  df-fn 4892  df-f 4893  df-f1 4894  df-fo 4895  df-f1o 4896  df-fv 4897  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-1st 5754  df-2nd 5755  df-recs 5907  df-irdg 5944  df-1o 5988  df-2o 5989  df-oadd 5992  df-omul 5993  df-er 6093  df-ec 6095  df-qs 6099  df-ni 6383  df-pli 6384  df-mi 6385  df-lti 6386  df-plpq 6423  df-mpq 6424  df-enq 6426  df-nqqs 6427  df-plqqs 6428  df-mqqs 6429  df-1nqqs 6430  df-rq 6431  df-ltnqqs 6432  df-enq0 6503  df-nq0 6504  df-0nq0 6505  df-plq0 6506  df-mq0 6507  df-inp 6545  df-iplp 6547  df-iltp 6549
This theorem is referenced by: (None)
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