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Theorem caucvgprprlemexb 7483
Description: Lemma for caucvgprpr 7488. 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 7481 . . . . 5  |-  ( ph  ->  L  e.  P. )
6 caucvgprprlemexb.r . . . . . 6  |-  ( ph  ->  R  e.  N. )
7 recnnpr 7324 . . . . . 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 7313 . . . . 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 408 . . . 4  |-  ( ph  ->  ( L  +P.  <. { p  |  p  <Q  ( *Q `  [ <. R ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. R ,  1o >. ]  ~Q  )  <Q 
q } >. )  e.  P. )
111, 6ffvelrnd 5524 . . . 4  |-  ( ph  ->  ( F `  R
)  e.  P. )
12 caucvgprprlemexb.q . . . 4  |-  ( ph  ->  Q  e.  P. )
13 ltaprg 7395 . . . 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 1201 . . 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 7355 . . . . . 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 1201 . . . . 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 7354 . . . . . . 7  |-  ( ( Q  e.  P.  /\  L  e.  P. )  ->  ( Q  +P.  L
)  =  ( L  +P.  Q ) )
1812, 5, 17syl2anc 408 . . . . . 6  |-  ( ph  ->  ( Q  +P.  L
)  =  ( L  +P.  Q ) )
1918oveq1d 5757 . . . . 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 2152 . . . 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 7354 . . . . 5  |-  ( ( Q  e.  P.  /\  ( F `  R )  e.  P. )  -> 
( Q  +P.  ( F `  R )
)  =  ( ( F `  R )  +P.  Q ) )
2212, 11, 21syl2anc 408 . . . 4  |-  ( ph  ->  ( Q  +P.  ( F `  R )
)  =  ( ( F `  R )  +P.  Q ) )
2320, 22breq12d 3912 . . 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 187 . 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 274 . . . . 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 274 . . . . 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 274 . . . . 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 7198 . . . . . . 7  |-  ( R  e.  N.  ->  [ <. R ,  1o >. ]  ~Q  e.  Q. )
29 recclnq 7168 . . . . . . 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 274 . . . . 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 274 . . . . 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 109 . . . . 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 7482 . . . 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 7395 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
3635adantl 275 . . . . . . 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 5523 . . . . . . . . 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 7324 . . . . . . . . . 10  |-  ( b  e.  N.  ->  <. { p  |  p  <Q  ( *Q
`  [ <. b ,  1o >. ]  ~Q  ) } ,  { q  |  ( *Q `  [ <. b ,  1o >. ]  ~Q  )  <Q 
q } >.  e.  P. )
3938adantl 275 . . . . . . . . 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 7313 . . . . . . . . 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 408 . . . . . . . 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 479 . . . . . . . . 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 7313 . . . . . . . 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 408 . . . . . . 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 479 . . . . . . 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 479 . . . . . . 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 7354 . . . . . . . 8  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
4948adantl 275 . . . . . . 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 5908 . . . . . 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 7355 . . . . . . . 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 1201 . . . . . . 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 3909 . . . . . 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 7354 . . . . . . . . 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 408 . . . . . . . 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 5758 . . . . . . 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 3909 . . . . . 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 213 . . . . 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 2411 . . . 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 146 . . 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 114 . 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 169 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   1oc1o 6274   [cec 6395   N.cnpi 7048    <N clti 7051    ~Q ceq 7055   Q.cnq 7056    +Q cplq 7058   *Qcrq 7060    <Q cltq 7061   P.cnp 7067    +P. cpp 7069    <P cltp 7071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  caucvgprprlemaddq  7484
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