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Theorem cauappcvgprlemladd 6899
Description: Lemma for cauappcvgpr 6903. This takes  L and offsets it by the positive fraction  S. (Contributed by Jim Kingdon, 23-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f  |-  ( ph  ->  F : Q. --> Q. )
cauappcvgpr.app  |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
( F `  p
)  <Q  ( ( F `
 q )  +Q  ( p  +Q  q
) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  (
p  +Q  q ) ) ) )
cauappcvgpr.bnd  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
cauappcvgpr.lim  |-  L  = 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >.
cauappcvgprlemladd.s  |-  ( ph  ->  S  e.  Q. )
Assertion
Ref Expression
cauappcvgprlemladd  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
Distinct variable groups:    A, p    L, p, q    ph, p, q    F, l, u, p, q    S, l, q, u, p
Allowed substitution hints:    ph( u, l)    A( u, q, l)    L( u, l)

Proof of Theorem cauappcvgprlemladd
StepHypRef Expression
1 cauappcvgpr.f . . . 4  |-  ( ph  ->  F : Q. --> Q. )
2 cauappcvgpr.app . . . 4  |-  ( ph  ->  A. p  e.  Q.  A. q  e.  Q.  (
( F `  p
)  <Q  ( ( F `
 q )  +Q  ( p  +Q  q
) )  /\  ( F `  q )  <Q  ( ( F `  p )  +Q  (
p  +Q  q ) ) ) )
3 cauappcvgpr.bnd . . . 4  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
4 cauappcvgpr.lim . . . 4  |-  L  = 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u } >.
5 cauappcvgprlemladd.s . . . 4  |-  ( ph  ->  S  e.  Q. )
61, 2, 3, 4, 5cauappcvgprlemladdfl 6896 . . 3  |-  ( ph  ->  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  C_  ( 1st ` 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) )
71, 2, 3, 4, 5cauappcvgprlemladdrl 6898 . . 3  |-  ( ph  ->  ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) 
C_  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
86, 7eqssd 3017 . 2  |-  ( ph  ->  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  =  ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. ) )
91, 2, 3, 4, 5cauappcvgprlemladdfu 6895 . . 3  |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  C_  ( 2nd ` 
<. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) )
101, 2, 3, 4, 5cauappcvgprlemladdru 6897 . . 3  |-  ( ph  ->  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) 
C_  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )
119, 10eqssd 3017 . 2  |-  ( ph  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  =  ( 2nd `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. ) )
121, 2, 3, 4cauappcvgprlemcl 6894 . . . 4  |-  ( ph  ->  L  e.  P. )
13 nqprlu 6788 . . . . 5  |-  ( S  e.  Q.  ->  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  e.  P. )
145, 13syl 14 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )
15 addclpr 6778 . . . 4  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )  ->  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. )  e.  P. )
1612, 14, 15syl2anc 403 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )
17 npsspw 6712 . . . . . . 7  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
1817sseli 2996 . . . . . 6  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  ( ~P Q.  X.  ~P Q. ) )
19 1st2nd2 5826 . . . . . 6  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  ( ~P Q.  X.  ~P Q. )  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  <. ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) >. )
2018, 19syl 14 . . . . 5  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  <. ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) >. )
21 ssrab2 3080 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  C_  Q.
22 nqex 6604 . . . . . . . . 9  |-  Q.  e.  _V
2322elpw2 3934 . . . . . . . 8  |-  ( { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) }  e.  ~P Q.  <->  { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) }  C_  Q. )
2421, 23mpbir 144 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  ~P Q.
25 ssrab2 3080 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  C_  Q.
2622elpw2 3934 . . . . . . . 8  |-  ( { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u }  e.  ~P Q.  <->  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u }  C_  Q. )
2725, 26mpbir 144 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  ~P Q.
28 opelxpi 4396 . . . . . . 7  |-  ( ( { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) }  e.  ~P Q.  /\  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  ~P Q. )  ->  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >.  e.  ( ~P Q.  X.  ~P Q. ) )
2924, 27, 28mp2an 417 . . . . . 6  |-  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >.  e.  ( ~P Q.  X.  ~P Q. )
30 1st2nd2 5826 . . . . . 6  |-  ( <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >.  e.  ( ~P Q.  X.  ~P Q. )  ->  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >.  = 
<. ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ,  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
>. )
3129, 30mp1i 10 . . . . 5  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >.  = 
<. ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ,  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
>. )
3220, 31eqeq12d 2096 . . . 4  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. 
<-> 
<. ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) >.  =  <. ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ,  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
>. ) )
33 xp1st 5817 . . . . . 6  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  ( ~P Q.  X.  ~P Q. )  ->  ( 1st `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) )  e. 
~P Q. )
3418, 33syl 14 . . . . 5  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  e.  ~P Q. )
35 xp2nd 5818 . . . . . 6  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  ( ~P Q.  X.  ~P Q. )  ->  ( 2nd `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) )  e. 
~P Q. )
3618, 35syl 14 . . . . 5  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  e.  ~P Q. )
37 opthg 3995 . . . . 5  |-  ( ( ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  e.  ~P Q.  /\  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  e.  ~P Q. )  ->  ( <. ( 1st `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) ) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) >.  =  <. ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. ) ,  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
>. 
<->  ( ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  =  ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. )  /\  ( 2nd `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) )  =  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ) ) )
3834, 36, 37syl2anc 403 . . . 4  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( <. ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) ,  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) >.  =  <. ( 1st `  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ,  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
>. 
<->  ( ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  =  ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. )  /\  ( 2nd `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) )  =  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ) ) )
3932, 38bitrd 186 . . 3  |-  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P.  ->  ( ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. 
<->  ( ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  =  ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. )  /\  ( 2nd `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) )  =  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ) ) )
4016, 39syl 14 . 2  |-  ( ph  ->  ( ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. )  =  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >.  <->  (
( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  =  ( 1st `  <. { l  e. 
Q.  |  E. q  e.  Q.  ( l  +Q  q )  <Q  (
( F `  q
)  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u } >. )  /\  ( 2nd `  ( L  +P.  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >. ) )  =  ( 2nd `  <. { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) } ,  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. ) ) ) )
418, 11, 40mpbir2and 886 1  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  <. { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) } ,  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u } >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   {crab 2353    C_ wss 2974   ~Pcpw 3384   <.cop 3403   class class class wbr 3787    X. cxp 4363   -->wf 4922   ` cfv 4926  (class class class)co 5537   1stc1st 5790   2ndc2nd 5791   Q.cnq 6521    +Q cplq 6523    <Q cltq 6526   P.cnp 6532    +P. cpp 6534
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-iplp 6709  df-iltp 6711
This theorem is referenced by:  cauappcvgprlem1  6900
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