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Theorem caucvgsrlemgt1 7825
Description: Lemma for caucvgsr 7832. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
Hypotheses
Ref Expression
caucvgsr.f  |-  ( ph  ->  F : N. --> R. )
caucvgsr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
caucvgsrlemgt1.gt1  |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )
Assertion
Ref Expression
caucvgsrlemgt1  |-  ( ph  ->  E. y  e.  R.  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e. 
N.  ( j  <N 
i  ->  ( ( F `  i )  <R  ( y  +R  x
)  /\  y  <R  ( ( F `  i
)  +R  x ) ) ) ) )
Distinct variable groups:    j, F, k, l, u    i, F, x, j, k    m, F, n, k    n, l, u    y, F, i, j, x    ph, j,
k, x    ph, n    k, m, n
Allowed substitution hints:    ph( y, u, i, m, l)

Proof of Theorem caucvgsrlemgt1
Dummy variables  a  b  w  z  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgsr.f . . . 4  |-  ( ph  ->  F : N. --> R. )
2 caucvgsr.cau . . . 4  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <R  ( ( F `
 k )  +R 
[ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  (
( F `  n
)  +R  [ <. (
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ) ) ) )
3 caucvgsrlemgt1.gt1 . . . 4  |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )
4 eqid 2189 . . . 4  |-  ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) )  =  ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
)
51, 2, 3, 4caucvgsrlemf 7822 . . 3  |-  ( ph  ->  ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) : N. --> P. )
61, 2, 3, 4caucvgsrlemcau 7823 . . 3  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  n
)  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  n )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
71, 2, 3, 4caucvgsrlembound 7824 . . 3  |-  ( ph  ->  A. m  e.  N.  1P  <P  ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  m ) )
85, 6, 7caucvgprpr 7742 . 2  |-  ( ph  ->  E. a  e.  P.  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) )
9 prsrcl 7814 . . . 4  |-  ( a  e.  P.  ->  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
109ad2antrl 490 . . 3  |-  ( (
ph  /\  ( a  e.  P.  /\  A. b  e.  P.  E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) ) ) ) )  ->  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
11 oveq2 5905 . . . . . . . . . . . 12  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  (
a  +P.  b )  =  ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) )
1211breq2d 4030 . . . . . . . . . . 11  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  <->  ( (
z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) )
13 oveq2 5905 . . . . . . . . . . . 12  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )  =  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) )
1413breq2d 4030 . . . . . . . . . . 11  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  (
a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b )  <->  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) )
1512, 14anbi12d 473 . . . . . . . . . 10  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  (
( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) )  <->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) ) )
1615imbi2d 230 . . . . . . . . 9  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  (
( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) ) )  <-> 
( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) ) ) )
1716rexralbidv 2516 . . . . . . . 8  |-  ( b  =  ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  ->  ( E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) )  <->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) ) ) )
18 simplrr 536 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  P.  /\  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) ) )  /\  x  e.  R. )  ->  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) )
1918adantr 276 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  A. b  e.  P.  E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) ) ) )
20 srpospr 7813 . . . . . . . . . 10  |-  ( ( x  e.  R.  /\  0R  <R  x )  ->  E! c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )
21 riotacl 5867 . . . . . . . . . 10  |-  ( E! c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x  ->  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )
2220, 21syl 14 . . . . . . . . 9  |-  ( ( x  e.  R.  /\  0R  <R  x )  -> 
( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )
2322adantll 476 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )
2417, 19, 23rspcdva 2861 . . . . . . 7  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) ) )
25 nfv 1539 . . . . . . . . . . 11  |-  F/ j
ph
26 nfv 1539 . . . . . . . . . . . 12  |-  F/ j  a  e.  P.
27 nfcv 2332 . . . . . . . . . . . . 13  |-  F/_ j P.
28 nfre1 2533 . . . . . . . . . . . . 13  |-  F/ j E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) )
2927, 28nfralya 2530 . . . . . . . . . . . 12  |-  F/ j A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) )
3026, 29nfan 1576 . . . . . . . . . . 11  |-  F/ j ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) )
3125, 30nfan 1576 . . . . . . . . . 10  |-  F/ j ( ph  /\  (
a  e.  P.  /\  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) ) )
32 nfv 1539 . . . . . . . . . 10  |-  F/ j  x  e.  R.
3331, 32nfan 1576 . . . . . . . . 9  |-  F/ j ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )
34 nfv 1539 . . . . . . . . 9  |-  F/ j 0R  <R  x
3533, 34nfan 1576 . . . . . . . 8  |-  F/ j ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )
36 nfv 1539 . . . . . . . . . . . 12  |-  F/ k
ph
37 nfv 1539 . . . . . . . . . . . . 13  |-  F/ k  a  e.  P.
38 nfcv 2332 . . . . . . . . . . . . . 14  |-  F/_ k P.
39 nfcv 2332 . . . . . . . . . . . . . . 15  |-  F/_ k N.
40 nfra1 2521 . . . . . . . . . . . . . . 15  |-  F/ k A. k  e.  N.  ( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) ) )
4139, 40nfrexya 2531 . . . . . . . . . . . . . 14  |-  F/ k E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) )
4238, 41nfralya 2530 . . . . . . . . . . . . 13  |-  F/ k A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) )
4337, 42nfan 1576 . . . . . . . . . . . 12  |-  F/ k ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) )
4436, 43nfan 1576 . . . . . . . . . . 11  |-  F/ k ( ph  /\  (
a  e.  P.  /\  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) ) )
45 nfv 1539 . . . . . . . . . . 11  |-  F/ k  x  e.  R.
4644, 45nfan 1576 . . . . . . . . . 10  |-  F/ k ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )
47 nfv 1539 . . . . . . . . . 10  |-  F/ k 0R  <R  x
4846, 47nfan 1576 . . . . . . . . 9  |-  F/ k ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )
495ad4antr 494 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) : N. --> P. )
50 simpr 110 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  k  e.  N. )
5149, 50ffvelcdmd 5673 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  e.  P. )
52 simplrl 535 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
a  e.  P.  /\  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) ) )  /\  x  e.  R. )  ->  a  e.  P. )
5352adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  a  e.  P. )
54 addclpr 7567 . . . . . . . . . . . . . . 15  |-  ( ( a  e.  P.  /\  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )  ->  ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )
5553, 23, 54syl2anc 411 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  (
a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )
5655adantr 276 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )
57 prsrlt 7817 . . . . . . . . . . . . 13  |-  ( ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  e.  P.  /\  ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  <->  [ <. (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  1P ) ,  1P >. ]  ~R  <R  [
<. ( ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  ) )
5851, 56, 57syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  <->  [ <. (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  1P ) ,  1P >. ]  ~R  <R  [
<. ( ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  ) )
591, 2, 3, 4caucvgsrlemfv 7821 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  N. )  ->  [ <. ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
6059adantlr 477 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
a  e.  P.  /\  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) ) )  /\  k  e.  N. )  ->  [ <. (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
6160adantlr 477 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  k  e.  N. )  ->  [ <. ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
6261adantlr 477 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  [ <. ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  1P ) ,  1P >. ]  ~R  =  ( F `  k ) )
63 prsradd 7816 . . . . . . . . . . . . . . . 16  |-  ( ( a  e.  P.  /\  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )  ->  [ <. (
( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  ) )
6453, 23, 63syl2anc 411 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  [ <. ( ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  ) )
65 prsrriota 7818 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  R.  /\  0R  <R  x )  ->  [ <. ( ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  =  x )
6665oveq2d 5913 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  R.  /\  0R  <R  x )  -> 
( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  [
<. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  )  =  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x ) )
6766adantll 476 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  )  =  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x ) )
6864, 67eqtrd 2222 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  [ <. ( ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) )
6968adantr 276 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  [ <. ( ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) )
7062, 69breq12d 4031 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  ( [ <. ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  1P ) ,  1P >. ]  ~R  <R  [ <. ( ( a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  <->  ( F `  k )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) ) )
7158, 70bitrd 188 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  <->  ( F `  k )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) ) )
7253adantr 276 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  a  e.  P. )
7323adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )
74 addclpr 7567 . . . . . . . . . . . . . 14  |-  ( ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  e.  P.  /\  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )  ->  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )
7551, 73, 74syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )
76 prsrlt 7817 . . . . . . . . . . . . 13  |-  ( ( a  e.  P.  /\  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  e. 
P. )  ->  (
a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  <->  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  [
<. ( ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  ) )
7772, 75, 76syl2anc 411 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  <->  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  [
<. ( ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  ) )
78 prsradd 7816 . . . . . . . . . . . . . 14  |-  ( ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  e.  P.  /\  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  e.  P. )  ->  [ <. (
( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  ) )
7951, 73, 78syl2anc 411 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  [ <. ( ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  =  ( [ <. ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  ) )
8079breq2d 4030 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  <R  [ <. ( ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  +P. 
1P ) ,  1P >. ]  ~R  <->  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( [ <. ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  ) ) )
8165adantll 476 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  [ <. ( ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  =  x )
8281adantr 276 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  [ <. ( ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  =  x )
8362, 82oveq12d 5915 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  ( [ <. ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  )  =  ( ( F `  k )  +R  x ) )
8483breq2d 4030 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  <R  ( [ <. ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  1P ) ,  1P >. ]  ~R  +R  [ <. ( ( iota_ c  e. 
P.  [ <. (
c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  ) 
<->  [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  <R  (
( F `  k
)  +R  x ) ) )
8577, 80, 843bitrd 214 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  <->  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k
)  +R  x ) ) )
8671, 85anbi12d 473 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) )  <-> 
( ( F `  k )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
)  /\  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k
)  +R  x ) ) ) )
8786imbi2d 230 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  /\  k  e.  N. )  ->  (
( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) )  <->  ( j  <N 
k  ->  ( ( F `  k )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k )  +R  x
) ) ) ) )
8848, 87ralbida 2484 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  ( A. k  e.  N.  ( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) )  <->  A. k  e.  N.  ( j  <N  k  ->  ( ( F `  k )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
)  /\  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k
)  +R  x ) ) ) ) )
8935, 88rexbid 2489 . . . . . . 7  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  ( E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) )  /\  a  <P  ( ( ( z  e.  N.  |->  (
iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  ( iota_ c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x ) ) ) )  <->  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( F `  k
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k )  +R  x
) ) ) ) )
9024, 89mpbid 147 . . . . . 6  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  E. j  e.  N.  A. k  e. 
N.  ( j  <N 
k  ->  ( ( F `  k )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k )  +R  x
) ) ) )
91 breq2 4022 . . . . . . . . 9  |-  ( k  =  i  ->  (
j  <N  k  <->  j  <N  i ) )
92 fveq2 5534 . . . . . . . . . . 11  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
9392breq1d 4028 . . . . . . . . . 10  |-  ( k  =  i  ->  (
( F `  k
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  <->  ( F `  i )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) ) )
9492oveq1d 5912 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( F `  k
)  +R  x )  =  ( ( F `
 i )  +R  x ) )
9594breq2d 4030 . . . . . . . . . 10  |-  ( k  =  i  ->  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  <R  (
( F `  k
)  +R  x )  <->  [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  <R  (
( F `  i
)  +R  x ) ) )
9693, 95anbi12d 473 . . . . . . . . 9  |-  ( k  =  i  ->  (
( ( F `  k )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
)  /\  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k
)  +R  x ) )  <->  ( ( F `
 i )  <R 
( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) )
9791, 96imbi12d 234 . . . . . . . 8  |-  ( k  =  i  ->  (
( j  <N  k  ->  ( ( F `  k )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
)  /\  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k
)  +R  x ) ) )  <->  ( j  <N  i  ->  ( ( F `  i )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) )
9897cbvralv 2718 . . . . . . 7  |-  ( A. k  e.  N.  (
j  <N  k  ->  (
( F `  k
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k )  +R  x
) ) )  <->  A. i  e.  N.  ( j  <N 
i  ->  ( ( F `  i )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) )
9998rexbii 2497 . . . . . 6  |-  ( E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( F `  k
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  k )  +R  x
) ) )  <->  E. j  e.  N.  A. i  e. 
N.  ( j  <N 
i  ->  ( ( F `  i )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) )
10090, 99sylib 122 . . . . 5  |-  ( ( ( ( ph  /\  ( a  e.  P.  /\ 
A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  (
j  <N  k  ->  (
( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  <P  ( a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  +P.  b
) ) ) ) )  /\  x  e. 
R. )  /\  0R  <R  x )  ->  E. j  e.  N.  A. i  e. 
N.  ( j  <N 
i  ->  ( ( F `  i )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) )
101100ex 115 . . . 4  |-  ( ( ( ph  /\  (
a  e.  P.  /\  A. b  e.  P.  E. j  e.  N.  A. k  e.  N.  ( j  <N 
k  ->  ( (
( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  <P  ( a  +P.  b
)  /\  a  <P  ( ( ( z  e. 
N.  |->  ( iota_ w  e. 
P.  ( F `  z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k
)  +P.  b )
) ) ) )  /\  x  e.  R. )  ->  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) )
102101ralrimiva 2563 . . 3  |-  ( (
ph  /\  ( a  e.  P.  /\  A. b  e.  P.  E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) ) ) ) )  ->  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) )
103 oveq1 5904 . . . . . . . . . 10  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( y  +R  x )  =  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x ) )
104103breq2d 4030 . . . . . . . . 9  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( F `  i
)  <R  ( y  +R  x )  <->  ( F `  i )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) ) )
105 breq1 4021 . . . . . . . . 9  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( y  <R  ( ( F `  i )  +R  x )  <->  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i
)  +R  x ) ) )
106104, 105anbi12d 473 . . . . . . . 8  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( ( F `  i )  <R  (
y  +R  x )  /\  y  <R  (
( F `  i
)  +R  x ) )  <->  ( ( F `
 i )  <R 
( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) )
107106imbi2d 230 . . . . . . 7  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( j  <N  i  ->  ( ( F `  i )  <R  (
y  +R  x )  /\  y  <R  (
( F `  i
)  +R  x ) ) )  <->  ( j  <N  i  ->  ( ( F `  i )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) )
108107rexralbidv 2516 . . . . . 6  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i )  +R  x
) ) )  <->  E. j  e.  N.  A. i  e. 
N.  ( j  <N 
i  ->  ( ( F `  i )  <R  ( [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) )
109108imbi2d 230 . . . . 5  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i )  +R  x
) ) ) )  <-> 
( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) ) )
110109ralbidv 2490 . . . 4  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i )  +R  x
) ) ) )  <->  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) ) )
111110rspcev 2856 . . 3  |-  ( ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  e.  R.  /\ 
A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  /\  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  <R  ( ( F `  i )  +R  x
) ) ) ) )  ->  E. y  e.  R.  A. x  e. 
R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i )  +R  x
) ) ) ) )
11210, 102, 111syl2anc 411 . 2  |-  ( (
ph  /\  ( a  e.  P.  /\  A. b  e.  P.  E. j  e. 
N.  A. k  e.  N.  ( j  <N  k  ->  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `
 z )  =  [ <. ( w  +P.  1P ) ,  1P >. ]  ~R  ) ) `  k )  <P  (
a  +P.  b )  /\  a  <P  ( ( ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) `  k )  +P.  b ) ) ) ) )  ->  E. y  e.  R.  A. x  e. 
R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e.  N.  (
j  <N  i  ->  (
( F `  i
)  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i )  +R  x
) ) ) ) )
1138, 112rexlimddv 2612 1  |-  ( ph  ->  E. y  e.  R.  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. i  e. 
N.  ( j  <N 
i  ->  ( ( F `  i )  <R  ( y  +R  x
)  /\  y  <R  ( ( F `  i
)  +R  x ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   E!wreu 2470   <.cop 3610   class class class wbr 4018    |-> cmpt 4079   -->wf 5231   ` cfv 5235   iota_crio 5851  (class class class)co 5897   1oc1o 6435   [cec 6558   N.cnpi 7302    <N clti 7305    ~Q ceq 7309   *Qcrq 7314    <Q cltq 7315   P.cnp 7321   1Pc1p 7322    +P. cpp 7323    <P cltp 7325    ~R cer 7326   R.cnr 7327   0Rc0r 7328   1Rc1r 7329    +R cplr 7331    <R cltr 7333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-enq0 7454  df-nq0 7455  df-0nq0 7456  df-plq0 7457  df-mq0 7458  df-inp 7496  df-i1p 7497  df-iplp 7498  df-iltp 7500  df-enr 7756  df-nr 7757  df-plr 7758  df-ltr 7760  df-0r 7761  df-1r 7762
This theorem is referenced by:  caucvgsrlemoffres  7830
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