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Theorem caucvgsrlemgt1 7284
Description: Lemma for caucvgsr 7291. 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 2085 . . . 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 7281 . . 3  |-  ( ph  ->  ( z  e.  N.  |->  ( iota_ w  e.  P.  ( F `  z )  =  [ <. (
w  +P.  1P ) ,  1P >. ]  ~R  )
) : N. --> P. )
61, 2, 3, 4caucvgsrlemcau 7282 . . 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 7283 . . 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 7215 . 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 7273 . . . 4  |-  ( a  e.  P.  ->  [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  e.  R. )
109ad2antrl 474 . . 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 5621 . . . . . . . . . . . 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 3832 . . . . . . . . . . 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 5621 . . . . . . . . . . . 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 3832 . . . . . . . . . . 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 457 . . . . . . . . . 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 228 . . . . . . . . 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 2400 . . . . . . . 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 503 . . . . . . . . 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 270 . . . . . . . 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 7272 . . . . . . . . . 10  |-  ( ( x  e.  R.  /\  0R  <R  x )  ->  E! c  e.  P.  [
<. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )
21 riotacl 5583 . . . . . . . . . 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 460 . . . . . . . 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 2720 . . . . . . 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 1464 . . . . . . . . . . 11  |-  F/ j
ph
26 nfv 1464 . . . . . . . . . . . 12  |-  F/ j  a  e.  P.
27 nfcv 2225 . . . . . . . . . . . . 13  |-  F/_ j P.
28 nfre1 2415 . . . . . . . . . . . . 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 2412 . . . . . . . . . . . 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 1500 . . . . . . . . . . 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 1500 . . . . . . . . . 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 1464 . . . . . . . . . 10  |-  F/ j  x  e.  R.
3331, 32nfan 1500 . . . . . . . . 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 1464 . . . . . . . . 9  |-  F/ j 0R  <R  x
3533, 34nfan 1500 . . . . . . . 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 1464 . . . . . . . . . . . 12  |-  F/ k
ph
37 nfv 1464 . . . . . . . . . . . . 13  |-  F/ k  a  e.  P.
38 nfcv 2225 . . . . . . . . . . . . . 14  |-  F/_ k P.
39 nfcv 2225 . . . . . . . . . . . . . . 15  |-  F/_ k N.
40 nfra1 2405 . . . . . . . . . . . . . . 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 2413 . . . . . . . . . . . . . 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 2412 . . . . . . . . . . . . 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 1500 . . . . . . . . . . . 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 1500 . . . . . . . . . . 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 1464 . . . . . . . . . . 11  |-  F/ k  x  e.  R.
4644, 45nfan 1500 . . . . . . . . . 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 1464 . . . . . . . . . 10  |-  F/ k 0R  <R  x
4846, 47nfan 1500 . . . . . . . . 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 478 . . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . 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, 50ffvelrnd 5398 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 7040 . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . 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 7276 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 7280 . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 7275 . . . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . . . 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 7277 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  R.  /\  0R  <R  x )  ->  [ <. ( ( iota_ c  e.  P.  [ <. ( c  +P.  1P ) ,  1P >. ]  ~R  =  x )  +P.  1P ) ,  1P >. ]  ~R  =  x )
6665oveq2d 5629 . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . 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 2117 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . 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 3833 . . . . . . . . . . . 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 186 . . . . . . . . . . 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 270 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 7040 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 7276 . . . . . . . . . . . . 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 403 . . . . . . . . . . . 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 7275 . . . . . . . . . . . . . 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 403 . . . . . . . . . . . . 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 3832 . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 5631 . . . . . . . . . . . . 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 3832 . . . . . . . . . . . 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 212 . . . . . . . . . . 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 457 . . . . . . . . . 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 228 . . . . . . . . 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 2370 . . . . . . . 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 2375 . . . . . . 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 145 . . . . . 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 3824 . . . . . . . . 9  |-  ( k  =  i  ->  (
j  <N  k  <->  j  <N  i ) )
92 fveq2 5268 . . . . . . . . . . 11  |-  ( k  =  i  ->  ( F `  k )  =  ( F `  i ) )
9392breq1d 3830 . . . . . . . . . 10  |-  ( k  =  i  ->  (
( F `  k
)  <R  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x )  <->  ( F `  i )  <R  ( [ <. ( a  +P. 
1P ) ,  1P >. ]  ~R  +R  x
) ) )
9492oveq1d 5628 . . . . . . . . . . 11  |-  ( k  =  i  ->  (
( F `  k
)  +R  x )  =  ( ( F `
 i )  +R  x ) )
9594breq2d 3832 . . . . . . . . . 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 457 . . . . . . . . 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 232 . . . . . . . 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 2586 . . . . . . 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 2381 . . . . . 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 120 . . . . 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 113 . . . 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 2442 . . 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 5620 . . . . . . . . . 10  |-  ( y  =  [ <. (
a  +P.  1P ) ,  1P >. ]  ~R  ->  ( y  +R  x )  =  ( [ <. ( a  +P.  1P ) ,  1P >. ]  ~R  +R  x ) )
104103breq2d 3832 . . . . . . . . 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 3823 . . . . . . . . 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 457 . . . . . . . 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 228 . . . . . . 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 2400 . . . . . 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 228 . . . . 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 2376 . . . 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 2715 . . 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 403 . 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 2489 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 102    <-> wb 103    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   E.wrex 2356   E!wreu 2357   <.cop 3434   class class class wbr 3820    |-> cmpt 3874   -->wf 4977   ` cfv 4981   iota_crio 5568  (class class class)co 5613   1oc1o 6128   [cec 6242   N.cnpi 6775    <N clti 6778    ~Q ceq 6782   *Qcrq 6787    <Q cltq 6788   P.cnp 6794   1Pc1p 6795    +P. cpp 6796    <P cltp 6798    ~R cer 6799   R.cnr 6800   0Rc0r 6801   1Rc1r 6802    +R cplr 6804    <R cltr 6806
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-2o 6136  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856  df-enq0 6927  df-nq0 6928  df-0nq0 6929  df-plq0 6930  df-mq0 6931  df-inp 6969  df-i1p 6970  df-iplp 6971  df-iltp 6973  df-enr 7216  df-nr 7217  df-plr 7218  df-ltr 7220  df-0r 7221  df-1r 7222
This theorem is referenced by:  caucvgsrlemoffres  7289
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