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Theorem cauappcvgprlemladdfl 6627
Description: Lemma for cauappcvgprlemladd 6630. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f  F : Q. --> Q.
cauappcvgpr.app  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `
 p  +Q  p  +Q  q
cauappcvgpr.bnd  p  Q.  <Q  F `  p
cauappcvgpr.lim  L 
<. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.
cauappcvgprlemladd.s  S  Q.
Assertion
Ref Expression
cauappcvgprlemladdfl  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  1st `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
Distinct variable groups:   , p    L, p, q   , p, q    F, l,, p, q    S, l, q,
Allowed substitution hints:   (, l)   (, q, l)    S( p)    L(, l)

Proof of Theorem cauappcvgprlemladdfl
Dummy variables  h  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . . 7  F : Q. --> Q.
2 cauappcvgpr.app . . . . . . 7  p  Q.  q  Q.  F `  p 
<Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `
 p  +Q  p  +Q  q
3 cauappcvgpr.bnd . . . . . . 7  p  Q.  <Q  F `  p
4 cauappcvgpr.lim . . . . . . 7  L 
<. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.
51, 2, 3, 4cauappcvgprlemcl 6625 . . . . . 6  L  P.
6 cauappcvgprlemladd.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, 10genpelvl 6495 . . . . . 6  L  P.  <. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  P.  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
r  s  +Q  t
125, 8, 11syl2anc 391 . . . . 5  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
r  s  +Q  t
1312biimpa 280 . . . 4  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
r  s  +Q  t
14 oveq1 5462 . . . . . . . . . . . . . . . 16  l  s 
l  +Q  q  s  +Q  q
1514breq1d 3765 . . . . . . . . . . . . . . 15  l  s  l  +Q  q  <Q  F `  q  s  +Q  q  <Q  F `  q
1615rexbidv 2321 . . . . . . . . . . . . . 14  l  s  q  Q.  l  +Q  q  <Q  F `  q  q  Q.  s  +Q  q  <Q  F `  q
174fveq2i 5124 . . . . . . . . . . . . . . 15  1st `  L  1st `  <. { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.
18 nqex 6347 . . . . . . . . . . . . . . . . 17  Q.  _V
1918rabex 3892 . . . . . . . . . . . . . . . 16  { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q }  _V
2018rabex 3892 . . . . . . . . . . . . . . . 16  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  }  _V
2119, 20op1st 5715 . . . . . . . . . . . . . . 15  1st `  <. { l 
Q.  |  q  Q.  l  +Q  q  <Q  F `  q } ,  {  Q.  |  q  Q.  F `
 q  +Q  q  <Q  } >.  { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q }
2217, 21eqtri 2057 . . . . . . . . . . . . . 14  1st `  L  {
l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q }
2316, 22elrab2 2694 . . . . . . . . . . . . 13  s  1st `  L  s  Q.  q  Q.  s  +Q  q  <Q  F `  q
2423biimpi 113 . . . . . . . . . . . 12  s  1st `  L  s 
Q.  q  Q.  s  +Q  q  <Q  F `  q
2524ad2antrl 459 . . . . . . . . . . 11  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  Q.  q  Q.  s  +Q  q  <Q  F `  q
2625adantr 261 . . . . . . . . . 10  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  Q.  q  Q. 
s  +Q  q 
<Q  F `  q
2726simpld 105 . . . . . . . . 9  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s 
Q.
28 vex 2554 . . . . . . . . . . . . . . 15  t 
_V
29 breq1 3758 . . . . . . . . . . . . . . 15  l  t 
l  <Q  S  t  <Q  S
30 ltnqex 6531 . . . . . . . . . . . . . . . 16  { l  |  l  <Q  S }  _V
31 gtnqex 6532 . . . . . . . . . . . . . . . 16  {  |  S  <Q  }  _V
3230, 31op1st 5715 . . . . . . . . . . . . . . 15  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  { l  |  l 
<Q  S }
3328, 29, 32elab2 2684 . . . . . . . . . . . . . 14  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  t  <Q  S
3433biimpi 113 . . . . . . . . . . . . 13  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  t  <Q  S
3534ad2antll 460 . . . . . . . . . . . 12  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  t  <Q  S
3635adantr 261 . . . . . . . . . . 11  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  t  <Q  S
37 ltrelnq 6349 . . . . . . . . . . . 12  <Q  C_  Q.  X.  Q.
3837brel 4335 . . . . . . . . . . 11  t 
<Q  S 
t  Q.  S  Q.
3936, 38syl 14 . . . . . . . . . 10  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  t  Q.  S  Q.
4039simpld 105 . . . . . . . . 9  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  t 
Q.
41 addclnq 6359 . . . . . . . . 9  s  Q.  t  Q.  s  +Q  t  Q.
4227, 40, 41syl2anc 391 . . . . . . . 8  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  +Q  t 
Q.
43 eleq1 2097 . . . . . . . . 9  r  s  +Q  t 
r  Q.  s  +Q  t 
Q.
4443adantl 262 . . . . . . . 8  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r  Q.  s  +Q  t 
Q.
4542, 44mpbird 156 . . . . . . 7  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r 
Q.
4626simprd 107 . . . . . . . 8  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q
4727ad2antrr 457 . . . . . . . . . . . . 13  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  s  Q.
48 simplr 482 . . . . . . . . . . . . 13  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  q  Q.
4940ad2antrr 457 . . . . . . . . . . . . 13  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  t  Q.
50 addcomnqg 6365 . . . . . . . . . . . . . 14  Q.  Q.  +Q  +Q
5150adantl 262 . . . . . . . . . . . . 13  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  Q.  Q.  +Q  +Q
52 addassnqg 6366 . . . . . . . . . . . . . 14  Q.  Q.  h  Q.  +Q  +Q  h  +Q  +Q  h
5352adantl 262 . . . . . . . . . . . . 13  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  Q.  Q.  h  Q.  +Q  +Q  h  +Q  +Q  h
5447, 48, 49, 51, 53caov32d 5623 . . . . . . . . . . . 12  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  s  +Q  q  +Q  t  s  +Q  t  +Q  q
55 simpr 103 . . . . . . . . . . . . . 14  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  +Q  q  <Q  F `  q  s  +Q  q  <Q  F `  q
5635ad2antrr 457 . . . . . . . . . . . . . 14  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  +Q  q  <Q  F `  q  t  <Q  S
5737brel 4335 . . . . . . . . . . . . . . 15  s  +Q  q 
<Q  F `  q  s  +Q  q 
Q.  F `  q  Q.
58 lt2addnq 6388 . . . . . . . . . . . . . . 15  s  +Q  q  Q.  F `  q  Q.  t  Q.  S  Q.  s  +Q  q 
<Q  F `  q  t  <Q  S  s  +Q  q  +Q  t  <Q  F `
 q  +Q  S
5957, 39, 58syl2anr 274 . . . . . . . . . . . . . 14  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  +Q  q  <Q  F `  q  s  +Q  q  <Q  F `  q  t  <Q  S  s  +Q  q  +Q  t  <Q  F `  q  +Q  S
6055, 56, 59mp2and 409 . . . . . . . . . . . . 13  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  s  +Q  q  <Q  F `  q  s  +Q  q  +Q  t  <Q  F `  q  +Q  S
6160adantlr 446 . . . . . . . . . . . 12  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  s  +Q  q  +Q  t  <Q  F `  q  +Q  S
6254, 61eqbrtrrd 3777 . . . . . . . . . . 11  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  s  +Q  t  +Q  q  <Q  F `  q  +Q  S
63 oveq1 5462 . . . . . . . . . . . . 13  r  s  +Q  t 
r  +Q  q  s  +Q  t  +Q  q
6463breq1d 3765 . . . . . . . . . . . 12  r  s  +Q  t  r  +Q  q  <Q  F `
 q  +Q  S 
s  +Q  t  +Q  q  <Q  F `  q  +Q  S
6564ad3antlr 462 . . . . . . . . . . 11  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  r  +Q  q  <Q  F `
 q  +Q  S 
s  +Q  t  +Q  q  <Q  F `  q  +Q  S
6662, 65mpbird 156 . . . . . . . . . 10  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  s  +Q  t  q  Q.  s  +Q  q  <Q  F `  q  r  +Q  q  <Q  F `
 q  +Q  S
6766ex 108 . . . . . . . . 9  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  q 
Q.  s  +Q  q  <Q  F `  q 
r  +Q  q 
<Q  F `  q  +Q  S
6867reximdva 2415 . . . . . . . 8  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  q  Q. 
s  +Q  q 
<Q  F `  q  q 
Q.  r  +Q  q  <Q  F `  q  +Q  S
6946, 68mpd 13 . . . . . . 7  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  q  Q.  r  +Q  q  <Q  F `  q  +Q  S
70 oveq1 5462 . . . . . . . . . 10  l  r 
l  +Q  q  r  +Q  q
7170breq1d 3765 . . . . . . . . 9  l  r  l  +Q  q  <Q  F `
 q  +Q  S  r  +Q  q  <Q  F `  q  +Q  S
7271rexbidv 2321 . . . . . . . 8  l  r  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S  q  Q.  r  +Q  q  <Q  F `  q  +Q  S
7318rabex 3892 . . . . . . . . 9  { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q  +Q  S }  _V
7418rabex 3892 . . . . . . . . 9  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  }  _V
7573, 74op1st 5715 . . . . . . . 8  1st `  <. { l 
Q.  |  q  Q.  l  +Q  q  <Q  F `  q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.  {
l  Q.  |  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S }
7672, 75elrab2 2694 . . . . . . 7  r  1st `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.  r  Q.  q  Q. 
r  +Q  q 
<Q  F `  q  +Q  S
7745, 69, 76sylanbrc 394 . . . . . 6  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L  t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r  1st `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
7877ex 108 . . . . 5  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
s  1st `  L  t  1st `  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r 
s  +Q  t  r  1st `  <. { l 
Q.  |  q  Q.  l  +Q  q  <Q  F `  q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
7978rexlimdvva 2434 . . . 4  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  s  1st `  L t  1st ` 
<. { l  |  l 
<Q  S } ,  {  |  S  <Q  } >. r  s  +Q  t  r  1st `  <. { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
8013, 79mpd 13 . . 3  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  1st `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
8180ex 108 . 2  r  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  r  1st `  <. { l  Q.  |  q  Q. 
l  +Q  q 
<Q  F `  q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
8281ssrdv 2945 1  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  1st `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
 q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  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   1stc1st 5707   Q.cnq 6264    +Q cplq 6266    <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:  cauappcvgprlemladdru  6628  cauappcvgprlemladd  6630
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