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Theorem cauappcvgprlemladd 7480
Description: Lemma for cauappcvgpr 7484. 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 7477 . . 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 7479 . . 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 3114 . 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 7476 . . 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 7478 . . 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 3114 . 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 7475 . . . 4  |-  ( ph  ->  L  e.  P. )
13 nqprlu 7369 . . . . 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 7359 . . . 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 7293 . . . . . . 7  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
1817sseli 3093 . . . . . 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 6073 . . . . . 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 3182 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  C_  Q.
22 nqex 7185 . . . . . . . . 9  |-  Q.  e.  _V
2322elpw2 4082 . . . . . . . 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 3182 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  C_  Q.
2622elpw2 4082 . . . . . . . 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 4571 . . . . . . 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 6073 . . . . . 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 2154 . . . 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 6063 . . . . . 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 6064 . . . . . 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 4160 . . . . 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 928 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 1331    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420    C_ wss 3071   ~Pcpw 3510   <.cop 3530   class class class wbr 3929    X. cxp 4537   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7102    +Q cplq 7104    <Q cltq 7107   P.cnp 7113    +P. cpp 7115
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-enq0 7246  df-nq0 7247  df-0nq0 7248  df-plq0 7249  df-mq0 7250  df-inp 7288  df-iplp 7290  df-iltp 7292
This theorem is referenced by:  cauappcvgprlem1  7481
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