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Theorem cauappcvgprlemladd 7645
Description: Lemma for cauappcvgpr 7649. 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 7642 . . 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 7644 . . 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 3172 . 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 7641 . . 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 7643 . . 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 3172 . 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 7640 . . . 4  |-  ( ph  ->  L  e.  P. )
13 nqprlu 7534 . . . . 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 7524 . . . 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 411 . . 3  |-  ( ph  ->  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  e.  P. )
17 npsspw 7458 . . . . . . 7  |-  P.  C_  ( ~P Q.  X.  ~P Q. )
1817sseli 3151 . . . . . 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 6170 . . . . . 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 3240 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  C_  Q.
22 nqex 7350 . . . . . . . . 9  |-  Q.  e.  _V
2322elpw2 4154 . . . . . . . 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 146 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  ~P Q.
25 ssrab2 3240 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  C_  Q.
2622elpw2 4154 . . . . . . . 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 146 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  ~P Q.
28 opelxpi 4655 . . . . . . 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 426 . . . . . 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 6170 . . . . . 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 2192 . . . 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 6160 . . . . . 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 6161 . . . . . 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 4235 . . . . 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 411 . . . 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 188 . . 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 944 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459    C_ wss 3129   ~Pcpw 3574   <.cop 3594   class class class wbr 4000    X. cxp 4621   -->wf 5208   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7267    +Q cplq 7269    <Q cltq 7272   P.cnp 7278    +P. cpp 7280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7291  df-pli 7292  df-mi 7293  df-lti 7294  df-plpq 7331  df-mpq 7332  df-enq 7334  df-nqqs 7335  df-plqqs 7336  df-mqqs 7337  df-1nqqs 7338  df-rq 7339  df-ltnqqs 7340  df-enq0 7411  df-nq0 7412  df-0nq0 7413  df-plq0 7414  df-mq0 7415  df-inp 7453  df-iplp 7455  df-iltp 7457
This theorem is referenced by:  cauappcvgprlem1  7646
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