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Theorem cauappcvgprlemladd 7434
Description: Lemma for cauappcvgpr 7438. 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 7431 . . 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 7433 . . 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 3084 . 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 7430 . . 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 7432 . . 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 3084 . 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 7429 . . . 4  |-  ( ph  ->  L  e.  P. )
13 nqprlu 7323 . . . . 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 7313 . . . 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 408 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )
17 npsspw 7247 . . . . . . 7  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
1817sseli 3063 . . . . . 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 6041 . . . . . 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 3152 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  C_  Q.
22 nqex 7139 . . . . . . . . 9  |-  Q.  e.  _V
2322elpw2 4052 . . . . . . . 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 145 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  ~P Q.
25 ssrab2 3152 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  C_  Q.
2622elpw2 4052 . . . . . . . 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 145 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  ~P Q.
28 opelxpi 4541 . . . . . . 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 422 . . . . . 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 6041 . . . . . 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 2132 . . . 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 6031 . . . . . 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 6032 . . . . . 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 4130 . . . . 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 408 . . . 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 187 . . 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 913 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   {cab 2103   A.wral 2393   E.wrex 2394   {crab 2397    C_ wss 3041   ~Pcpw 3480   <.cop 3500   class class class wbr 3899    X. cxp 4507   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061   P.cnp 7067    +P. cpp 7069
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:  cauappcvgprlem1  7435
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