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Theorem caucvgprlemladdfu 6648
Description: Lemma for caucvgpr 6653. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
Hypotheses
Ref Expression
caucvgpr.f  F : N. --> Q.
caucvgpr.cau  n  N.  k  N.  n  <N  k  F `  n 
<Q  F `  k  +Q  *Q `  <. n ,  1o >.  ~Q  F `  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q
caucvgpr.bnd  j  N.  <Q  F `  j
caucvgpr.lim  L 
<. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  } >.
caucvgprlemladd.s  S  Q.
Assertion
Ref Expression
caucvgprlemladdfu  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  { 
Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
Distinct variable groups:   , j    j, F,, l    n, F, k    k, L, j    S, l,, j    j,
k
Allowed substitution hints:   (, j, k, n, l)   (, k, n, l)    S( k, n)    L(, n, l)

Proof of Theorem caucvgprlemladdfu
Dummy variables  m  r  s  t  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.f . . . . . . 7  F : N. --> Q.
2 caucvgpr.cau . . . . . . 7  n  N.  k  N.  n  <N  k  F `  n 
<Q  F `  k  +Q  *Q `  <. n ,  1o >.  ~Q  F `  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q
3 caucvgpr.bnd . . . . . . 7  j  N.  <Q  F `  j
4 caucvgpr.lim . . . . . . 7  L 
<. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  } >.
51, 2, 3, 4caucvgprlemcl 6647 . . . . . 6  L  P.
6 caucvgprlemladd.s . . . . . . 7  S  Q.
7 nqprlu 6530 . . . . . . 7  S  Q.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  P.
86, 7syl 14 . . . . . 6  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  P.
9 df-iplp 6451 . . . . . . 7  +P.  P. ,  P.  |->  <. {  Q.  |  Q.  h  Q.  1st `  h  1st `  +Q  h } ,  {  Q.  |  Q.  h  Q.  2nd `  h  2nd `  +Q  h } >.
10 addclnq 6359 . . . . . . 7  Q.  h  Q.  +Q  h  Q.
119, 10genpelvu 6496 . . . . . 6  L  P.  <. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  P.  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
r  s  +Q  t
125, 8, 11syl2anc 391 . . . . 5  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
r  s  +Q  t
1312biimpa 280 . . . 4  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
r  s  +Q  t
14 breq2 3759 . . . . . . . . . . . . . . . 16  s  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  F `  j  +Q  *Q `  <. j ,  1o >. 
~Q  <Q  s
1514rexbidv 2321 . . . . . . . . . . . . . . 15  s  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  s
164fveq2i 5124 . . . . . . . . . . . . . . . 16  2nd `  L  2nd `  <. { l  Q.  |  j  N. 
l  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  F `  j } ,  {  Q.  |  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.
17 nqex 6347 . . . . . . . . . . . . . . . . . 18  Q.  _V
1817rabex 3892 . . . . . . . . . . . . . . . . 17  { l  Q.  |  j  N. 
l  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  F `  j }  _V
1917rabex 3892 . . . . . . . . . . . . . . . . 17  {  Q.  |  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  }  _V
2018, 19op2nd 5716 . . . . . . . . . . . . . . . 16  2nd `  <. { l 
Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >. 
~Q  <Q  F `  j } ,  {  Q.  |  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.  {  Q.  |  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  }
2116, 20eqtri 2057 . . . . . . . . . . . . . . 15  2nd `  L  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  }
2215, 21elrab2 2694 . . . . . . . . . . . . . 14  s  2nd `  L  s  Q.  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  s
2322biimpi 113 . . . . . . . . . . . . 13  s  2nd `  L  s 
Q.  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  s
2423adantr 261 . . . . . . . . . . . 12  s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  Q.  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  s
2524adantl 262 . . . . . . . . . . 11  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  Q.  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  s
2625adantr 261 . . . . . . . . . 10  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  Q.  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  s
2726simpld 105 . . . . . . . . 9  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s 
Q.
28 vex 2554 . . . . . . . . . . . . . 14  t 
_V
29 breq2 3759 . . . . . . . . . . . . . 14  t  S  <Q  S  <Q  t
30 ltnqex 6531 . . . . . . . . . . . . . . 15  { l  |  l  <Q  S }  _V
31 gtnqex 6532 . . . . . . . . . . . . . . 15  {  |  S  <Q  }  _V
3230, 31op2nd 5716 . . . . . . . . . . . . . 14  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  {  |  S  <Q  }
3328, 29, 32elab2 2684 . . . . . . . . . . . . 13  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  S  <Q  t
34 ltrelnq 6349 . . . . . . . . . . . . . 14  <Q  C_  Q.  X.  Q.
3534brel 4335 . . . . . . . . . . . . 13  S 
<Q  t  S  Q.  t  Q.
3633, 35sylbi 114 . . . . . . . . . . . 12  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  S  Q.  t  Q.
3736simprd 107 . . . . . . . . . . 11  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  t 
Q.
3837ad2antll 460 . . . . . . . . . 10  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  t  Q.
3938adantr 261 . . . . . . . . 9  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  t 
Q.
40 addclnq 6359 . . . . . . . . 9  s  Q.  t  Q.  s  +Q  t  Q.
4127, 39, 40syl2anc 391 . . . . . . . 8  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  +Q  t 
Q.
42 eleq1 2097 . . . . . . . . 9  r  s  +Q  t 
r  Q.  s  +Q  t 
Q.
4342adantl 262 . . . . . . . 8  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r  Q.  s  +Q  t 
Q.
4441, 43mpbird 156 . . . . . . 7  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r 
Q.
4526simprd 107 . . . . . . . . . 10  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  j  N.  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  s
46 fveq2 5121 . . . . . . . . . . . . 13  j  m  F `  j  F `  m
47 opeq1 3540 . . . . . . . . . . . . . . 15  j  m  <. j ,  1o >.  <. m ,  1o >.
4847eceq1d 6078 . . . . . . . . . . . . . 14  j  m  <. j ,  1o >. 
~Q  <. m ,  1o >.  ~Q
4948fveq2d 5125 . . . . . . . . . . . . 13  j  m  *Q `  <. j ,  1o >.  ~Q  *Q `  <. m ,  1o >. 
~Q
5046, 49oveq12d 5473 . . . . . . . . . . . 12  j  m  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  F `
 m  +Q  *Q `  <. m ,  1o >.  ~Q
5150breq1d 3765 . . . . . . . . . . 11  j  m  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  s  F `  m  +Q  *Q `  <. m ,  1o >. 
~Q  <Q  s
5251cbvrexv 2528 . . . . . . . . . 10  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
<Q  s  m  N.  F `
 m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s
5345, 52sylib 127 . . . . . . . . 9  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  m  N.  F `
 m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s
5433biimpi 113 . . . . . . . . . . . . . . . . 17  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  S  <Q  t
5554ad2antll 460 . . . . . . . . . . . . . . . 16  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
S  <Q  t
5655adantr 261 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  S  <Q  t
5756ad2antrr 457 . . . . . . . . . . . . . 14  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s 
S  <Q  t
586ad5antr 465 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s 
S  Q.
5939ad2antrr 457 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  t  Q.
601ad5antr 465 . . . . . . . . . . . . . . . . 17  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s 
F : N. --> Q.
61 simplr 482 . . . . . . . . . . . . . . . . 17  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s 
m  N.
6260, 61ffvelrnd 5246 . . . . . . . . . . . . . . . 16  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  Q.
63 nnnq 6405 . . . . . . . . . . . . . . . . 17  m  N.  <. m ,  1o >. 
~Q  Q.
64 recclnq 6376 . . . . . . . . . . . . . . . . 17  <. m ,  1o >. 
~Q  Q.  *Q `  <. m ,  1o >.  ~Q  Q.
6561, 63, 643syl 17 . . . . . . . . . . . . . . . 16  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  *Q `  <. m ,  1o >.  ~Q  Q.
66 addclnq 6359 . . . . . . . . . . . . . . . 16  F `  m  Q.  *Q `  <. m ,  1o >.  ~Q  Q.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  Q.
6762, 65, 66syl2anc 391 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  Q.
68 ltanqg 6384 . . . . . . . . . . . . . . 15  S  Q.  t  Q.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  Q.  S  <Q  t  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t
6958, 59, 67, 68syl3anc 1134 . . . . . . . . . . . . . 14  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  S  <Q  t  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t
7057, 69mpbid 135 . . . . . . . . . . . . 13  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t
71 simpr 103 . . . . . . . . . . . . . 14  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q 
<Q  s
72 ltanqg 6384 . . . . . . . . . . . . . . . 16  Q.  Q.  Q.  <Q  +Q  <Q  +Q
7372adantl 262 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  Q.  Q.  Q.  <Q  +Q  <Q  +Q
7427ad2antrr 457 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  s  Q.
75 addcomnqg 6365 . . . . . . . . . . . . . . . 16  Q.  Q.  +Q  +Q
7675adantl 262 . . . . . . . . . . . . . . 15  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  Q.  Q.  +Q  +Q
7773, 67, 74, 59, 76caovord2d 5612 . . . . . . . . . . . . . 14  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t  <Q  s  +Q  t
7871, 77mpbid 135 . . . . . . . . . . . . 13  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t  <Q  s  +Q  t
79 ltsonq 6382 . . . . . . . . . . . . . 14  <Q  Or  Q.
8079, 34sotri 4663 . . . . . . . . . . . . 13  F `
 m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  t  <Q  s  +Q  t  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  s  +Q  t
8170, 78, 80syl2anc 391 . . . . . . . . . . . 12  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  s  +Q  t
82 simpllr 486 . . . . . . . . . . . 12  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  r  s  +Q  t
8381, 82breqtrrd 3781 . . . . . . . . . . 11  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  <Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  r
8483ex 108 . . . . . . . . . 10  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  m 
N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q 
<Q  s  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  r
8584reximdva 2415 . . . . . . . . 9  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q 
<Q  s  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  r
8653, 85mpd 13 . . . . . . . 8  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  r
8750oveq1d 5470 . . . . . . . . . 10  j  m  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  F `
 m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S
8887breq1d 3765 . . . . . . . . 9  j  m  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  r  F `
 m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  r
8988cbvrexv 2528 . . . . . . . 8  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  r  m  N.  F `  m  +Q  *Q `  <. m ,  1o >.  ~Q  +Q  S  <Q  r
9086, 89sylibr 137 . . . . . . 7  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  r
91 breq2 3759 . . . . . . . . 9  r  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  F `
 j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  r
9291rexbidv 2321 . . . . . . . 8  r  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  r
9392elrab 2692 . . . . . . 7  r  { 
Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }  r 
Q.  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  r
9444, 90, 93sylanbrc 394 . . . . . 6  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L  t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r 
{  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
9594ex 108 . . . . 5  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  2nd `  L  t  2nd `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
9695rexlimdvva 2434 . . . 4  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  2nd `  L t  2nd ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >. r  s  +Q  t  r  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
9713, 96mpd 13 . . 3  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
9897ex 108 . 2  r  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
9998ssrdv 2945 1  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  { 
Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  +Q  S  <Q  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390   {cab 2023  wral 2300  wrex 2301   {crab 2304    C_ wss 2911   <.cop 3370   class class class wbr 3755   -->wf 4841   ` cfv 4845  (class class class)co 5455   2ndc2nd 5708   1oc1o 5933  cec 6040   N.cnpi 6256    <N clti 6259    ~Q ceq 6263   Q.cnq 6264    +Q cplq 6266   *Qcrq 6268    <Q cltq 6269   P.cnp 6275    +P. cpp 6277
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-iplp 6451
This theorem is referenced by:  caucvgprlemladdrl  6649
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