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Theorem caucvgprprlemexb 7245
Description: Lemma for caucvgprpr 7250. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)
Hypotheses
Ref Expression
caucvgprpr.f  |-  ( ph  ->  F : N. --> P. )
caucvgprpr.cau  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
caucvgprpr.bnd  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
caucvgprpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
caucvgprprlemexb.q  |-  ( ph  ->  Q  e.  P. )
caucvgprprlemexb.r  |-  ( ph  ->  R  e.  N. )
Assertion
Ref Expression
caucvgprprlemexb  |-  ( ph  ->  ( ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  R )  +P.  Q
)  ->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )  <P 
( ( F `  R )  +P.  Q
) ) )
Distinct variable groups:    A, m    m, F    A, r, m    F, b    k, F, l, n, u    F, r    L, b   
k, L    R, b, p, q    ph, b    k, p, q, r, l, u   
r, b
Allowed substitution hints:    ph( u, k, m, n, r, q, p, l)    A( u, k, n, q, p, b, l)    Q( u, k, m, n, r, q, p, b, l)    R( u, k, m, n, r, l)    F( q, p)    L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemexb
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . . . 6  |-  ( ph  ->  F : N. --> P. )
2 caucvgprpr.cau . . . . . 6  |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  (
n  <N  k  ->  (
( F `  n
)  <P  ( ( F `
 k )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
3 caucvgprpr.bnd . . . . . 6  |-  ( ph  ->  A. m  e.  N.  A  <P  ( F `  m ) )
4 caucvgprpr.lim . . . . . 6  |-  L  = 
<. { l  e.  Q.  |  E. r  e.  N.  <. { p  |  p  <Q  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  )
) } ,  {
q  |  ( l  +Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) ) 
<Q  q } >.  <P  ( F `  r ) } ,  { u  e.  Q.  |  E. r  e.  N.  ( ( F `
 r )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. r ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. r ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  <. { p  |  p  <Q  u } ,  { q  |  u 
<Q  q } >. } >.
51, 2, 3, 4caucvgprprlemclphr 7243 . . . . 5  |-  ( ph  ->  L  e.  P. )
6 caucvgprprlemexb.r . . . . . 6  |-  ( ph  ->  R  e.  N. )
7 recnnpr 7086 . . . . . 6  |-  ( R  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
86, 7syl 14 . . . . 5  |-  ( ph  -> 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )
9 addclpr 7075 . . . . 5  |-  ( ( L  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
105, 8, 9syl2anc 403 . . . 4  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
111, 6ffvelrnd 5419 . . . 4  |-  ( ph  ->  ( F `  R
)  e.  P. )
12 caucvgprprlemexb.q . . . 4  |-  ( ph  ->  Q  e.  P. )
13 ltaprg 7157 . . . 4  |-  ( ( ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  ( F `
 R )  e. 
P.  /\  Q  e.  P. )  ->  ( ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  R
)  <->  ( Q  +P.  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( Q  +P.  ( F `  R ) ) ) )
1410, 11, 12, 13syl3anc 1174 . . 3  |-  ( ph  ->  ( ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( F `  R
)  <->  ( Q  +P.  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( Q  +P.  ( F `  R ) ) ) )
15 addassprg 7117 . . . . . 6  |-  ( ( Q  e.  P.  /\  L  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )  ->  ( ( Q  +P.  L )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  =  ( Q  +P.  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
) )
1612, 5, 8, 15syl3anc 1174 . . . . 5  |-  ( ph  ->  ( ( Q  +P.  L )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( Q  +P.  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
) )
17 addcomprg 7116 . . . . . . 7  |-  ( ( Q  e.  P.  /\  L  e.  P. )  ->  ( Q  +P.  L
)  =  ( L  +P.  Q ) )
1812, 5, 17syl2anc 403 . . . . . 6  |-  ( ph  ->  ( Q  +P.  L
)  =  ( L  +P.  Q ) )
1918oveq1d 5649 . . . . 5  |-  ( ph  ->  ( ( Q  +P.  L )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  =  ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )
2016, 19eqtr3d 2122 . . . 4  |-  ( ph  ->  ( Q  +P.  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )  =  ( ( L  +P.  Q )  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
21 addcomprg 7116 . . . . 5  |-  ( ( Q  e.  P.  /\  ( F `  R )  e.  P. )  -> 
( Q  +P.  ( F `  R )
)  =  ( ( F `  R )  +P.  Q ) )
2212, 11, 21syl2anc 403 . . . 4  |-  ( ph  ->  ( Q  +P.  ( F `  R )
)  =  ( ( F `  R )  +P.  Q ) )
2320, 22breq12d 3850 . . 3  |-  ( ph  ->  ( ( Q  +P.  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( Q  +P.  ( F `  R ) )  <->  ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  R )  +P.  Q
) ) )
2414, 23bitrd 186 . 2  |-  ( ph  ->  ( ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( F `  R
)  <->  ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  R )  +P.  Q
) ) )
251adantr 270 . . . . 5  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  F : N.
--> P. )
262adantr 270 . . . . 5  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  A. n  e.  N.  A. k  e. 
N.  ( n  <N  k  ->  ( ( F `
 n )  <P 
( ( F `  k )  +P.  <. { l  |  l  <Q 
( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )  /\  ( F `  k
)  <P  ( ( F `
 n )  +P. 
<. { l  |  l 
<Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. )
) ) )
273adantr 270 . . . . 5  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  A. m  e.  N.  A  <P  ( F `  m )
)
28 nnnq 6960 . . . . . . 7  |-  ( R  e.  N.  ->  [ <. R ,  1o >. ]  ~Q  e.  Q. )
29 recclnq 6930 . . . . . . 7  |-  ( [
<. R ,  1o >. ]  ~Q  e.  Q.  ->  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  e.  Q. )
306, 28, 293syl 17 . . . . . 6  |-  ( ph  ->  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  e.  Q. )
3130adantr 270 . . . . 5  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  e. 
Q. )
3211adantr 270 . . . . 5  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  ( F `  R )  e.  P. )
33 simpr 108 . . . . 5  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)
3425, 26, 27, 4, 31, 32, 33caucvgprprlemexbt 7244 . . . 4  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  R
) )
35 ltaprg 7157 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
3635adantl 271 . . . . . . 7  |-  ( ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  R
) )  /\  b  e.  N. )  /\  (
f  e.  P.  /\  g  e.  P.  /\  h  e.  P. ) )  -> 
( f  <P  g  <->  ( h  +P.  f ) 
<P  ( h  +P.  g
) ) )
3725ffvelrnda 5418 . . . . . . . . 9  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( F `
 b )  e. 
P. )
38 recnnpr 7086 . . . . . . . . . 10  |-  ( b  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
3938adantl 271 . . . . . . . . 9  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
40 addclpr 7075 . . . . . . . . 9  |-  ( ( ( F `  b
)  e.  P.  /\  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P. )  ->  (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4137, 39, 40syl2anc 403 . . . . . . . 8  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
426ad2antrr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  R  e. 
N. )
4342, 7syl 14 . . . . . . . 8  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
44 addclpr 7075 . . . . . . . 8  |-  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )  ->  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
4541, 43, 44syl2anc 403 . . . . . . 7  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  e.  P. )
4611ad2antrr 472 . . . . . . 7  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( F `
 R )  e. 
P. )
4712ad2antrr 472 . . . . . . 7  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  Q  e. 
P. )
48 addcomprg 7116 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
4948adantl 271 . . . . . . 7  |-  ( ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  <P  ( F `  R
) )  /\  b  e.  N. )  /\  (
f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
5036, 45, 46, 47, 49caovord2d 5796 . . . . . 6  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )  <->  ( ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  Q
)  <P  ( ( F `
 R )  +P. 
Q ) ) )
51 addassprg 7117 . . . . . . . 8  |-  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P.  /\  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P.  /\  Q  e.  P. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  Q )  =  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P.  Q )
) )
5241, 43, 47, 51syl3anc 1174 . . . . . . 7  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  Q
)  =  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P.  Q )
) )
5352breq1d 3847 . . . . . 6  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( ( ( F `
 b )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  Q
)  <P  ( ( F `
 R )  +P. 
Q )  <->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P.  Q )
)  <P  ( ( F `
 R )  +P. 
Q ) ) )
54 addcomprg 7116 . . . . . . . . 9  |-  ( (
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  e. 
P.  /\  Q  e.  P. )  ->  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P. 
Q )  =  ( Q  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
5543, 47, 54syl2anc 403 . . . . . . . 8  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P. 
Q )  =  ( Q  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)
5655oveq2d 5650 . . . . . . 7  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P.  Q )
)  =  ( ( ( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
) )
5756breq1d 3847 . . . . . 6  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >.  +P.  Q )
)  <P  ( ( F `
 R )  +P. 
Q )  <->  ( (
( F `  b
)  +P.  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( ( F `
 R )  +P. 
Q ) ) )
5850, 53, 573bitrd 212 . . . . 5  |-  ( ( ( ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  /\  b  e.  N. )  ->  ( ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )  <->  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( ( F `
 R )  +P. 
Q ) ) )
5958rexbidva 2377 . . . 4  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  ( E. b  e.  N.  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )  <->  E. b  e.  N.  (
( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )
)  <P  ( ( F `
 R )  +P. 
Q ) ) )
6034, 59mpbid 145 . . 3  |-  ( (
ph  /\  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. )  <P  ( F `  R )
)  ->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )  <P 
( ( F `  R )  +P.  Q
) )
6160ex 113 . 2  |-  ( ph  ->  ( ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( F `  R
)  ->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )  <P 
( ( F `  R )  +P.  Q
) ) )
6224, 61sylbird 168 1  |-  ( ph  ->  ( ( ( L  +P.  Q )  +P. 
<. { p  |  p 
<Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) 
<P  ( ( F `  R )  +P.  Q
)  ->  E. b  e.  N.  ( ( ( F `  b )  +P.  <. { p  |  p  <Q  ( *Q `  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q  q } >. )  +P.  ( Q  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q  q } >. ) )  <P 
( ( F `  R )  +P.  Q
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1oc1o 6156   [cec 6270   N.cnpi 6810    <N clti 6813    ~Q ceq 6817   Q.cnq 6818    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823   P.cnp 6829    +P. cpp 6831    <P cltp 6833
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  caucvgprprlemaddq  7246
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