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Theorem cauappcvgprlemladdfl 7604
Description: Lemma for cauappcvgprlemladd 7607. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-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
cauappcvgprlemladdfl  |-  ( 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 } >. ) )
Distinct variable groups:    A, p    L, p, q    ph, p, q    F, l, u, p, q    S, l, q, u
Allowed substitution hints:    ph( u, l)    A( u, q, l)    S( p)    L( u, l)

Proof of Theorem cauappcvgprlemladdfl
Dummy variables  f  g  h  r  s  t  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . . . . 7  |-  ( ph  ->  F : Q. --> Q. )
2 cauappcvgpr.app . . . . . . 7  |-  ( 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 . . . . . . 7  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
4 cauappcvgpr.lim . . . . . . 7  |-  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 } >.
51, 2, 3, 4cauappcvgprlemcl 7602 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 cauappcvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 7496 . . . . . . 7  |-  ( S  e.  Q.  ->  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >.  e.  P. )
86, 7syl 14 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )
9 df-iplp 7417 . . . . . . 7  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
10 addclnq 7324 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvl 7461 . . . . . 6  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 1st `  L ) E. t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
125, 8, 11syl2anc 409 . . . . 5  |-  ( ph  ->  ( r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 1st `  L ) E. t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
1312biimpa 294 . . . 4  |-  ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  E. s  e.  ( 1st `  L
) E. t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) r  =  ( s  +Q  t ) )
14 oveq1 5857 . . . . . . . . . . . . . . . 16  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
1514breq1d 3997 . . . . . . . . . . . . . . 15  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
1615rexbidv 2471 . . . . . . . . . . . . . 14  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
174fveq2i 5497 . . . . . . . . . . . . . . 15  |-  ( 1st `  L )  =  ( 1st `  <. { 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 } >. )
18 nqex 7312 . . . . . . . . . . . . . . . . 17  |-  Q.  e.  _V
1918rabex 4131 . . . . . . . . . . . . . . . 16  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2018rabex 4131 . . . . . . . . . . . . . . . 16  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2119, 20op1st 6122 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. { 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 } >. )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
2217, 21eqtri 2191 . . . . . . . . . . . . . 14  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
2316, 22elrab2 2889 . . . . . . . . . . . . 13  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
2423biimpi 119 . . . . . . . . . . . 12  |-  ( s  e.  ( 1st `  L
)  ->  ( s  e.  Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
2524ad2antrl 487 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( s  e.  Q.  /\ 
E. q  e.  Q.  ( s  +Q  q
)  <Q  ( F `  q ) ) )
2625adantr 274 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  e.  Q.  /\ 
E. q  e.  Q.  ( s  +Q  q
)  <Q  ( F `  q ) ) )
2726simpld 111 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
s  e.  Q. )
28 vex 2733 . . . . . . . . . . . . . . 15  |-  t  e. 
_V
29 breq1 3990 . . . . . . . . . . . . . . 15  |-  ( l  =  t  ->  (
l  <Q  S  <->  t  <Q  S ) )
30 ltnqex 7498 . . . . . . . . . . . . . . . 16  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 7499 . . . . . . . . . . . . . . . 16  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op1st 6122 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { l  |  l 
<Q  S }
3328, 29, 32elab2 2878 . . . . . . . . . . . . . 14  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  t  <Q  S )
3433biimpi 119 . . . . . . . . . . . . 13  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  t  <Q  S )
3534ad2antll 488 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
t  <Q  S )
3635adantr 274 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
t  <Q  S )
37 ltrelnq 7314 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
3837brel 4661 . . . . . . . . . . 11  |-  ( t 
<Q  S  ->  ( t  e.  Q.  /\  S  e.  Q. ) )
3936, 38syl 14 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( t  e.  Q.  /\  S  e.  Q. )
)
4039simpld 111 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
t  e.  Q. )
41 addclnq 7324 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4227, 40, 41syl2anc 409 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  +Q  t
)  e.  Q. )
43 eleq1 2233 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4443adantl 275 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4542, 44mpbird 166 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  Q. )
4626simprd 113 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. q  e.  Q.  ( s  +Q  q
)  <Q  ( F `  q ) )
4727ad2antrr 485 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
s  e.  Q. )
48 simplr 525 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
q  e.  Q. )
4940ad2antrr 485 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
t  e.  Q. )
50 addcomnqg 7330 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5150adantl 275 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
52 addassnqg 7331 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5352adantl 275 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( (
f  +Q  g )  +Q  h )  =  ( f  +Q  (
g  +Q  h ) ) )
5447, 48, 49, 51, 53caov32d 6030 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
( ( s  +Q  q )  +Q  t
)  =  ( ( s  +Q  t )  +Q  q ) )
55 simpr 109 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
( s  +Q  q
)  <Q  ( F `  q ) )
5635ad2antrr 485 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
t  <Q  S )
5737brel 4661 . . . . . . . . . . . . . . 15  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  ->  ( (
s  +Q  q )  e.  Q.  /\  ( F `  q )  e.  Q. ) )
58 lt2addnq 7353 . . . . . . . . . . . . . . 15  |-  ( ( ( ( s  +Q  q )  e.  Q.  /\  ( F `  q
)  e.  Q. )  /\  ( t  e.  Q.  /\  S  e.  Q. )
)  ->  ( (
( s  +Q  q
)  <Q  ( F `  q )  /\  t  <Q  S )  ->  (
( s  +Q  q
)  +Q  t ) 
<Q  ( ( F `  q )  +Q  S
) ) )
5957, 39, 58syl2anr 288 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  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 431 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
( ( s  +Q  q )  +Q  t
)  <Q  ( ( F `
 q )  +Q  S ) )
6160adantlr 474 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
( ( s  +Q  q )  +Q  t
)  <Q  ( ( F `
 q )  +Q  S ) )
6254, 61eqbrtrrd 4011 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
( ( s  +Q  t )  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) )
63 oveq1 5857 . . . . . . . . . . . . 13  |-  ( r  =  ( s  +Q  t )  ->  (
r  +Q  q )  =  ( ( s  +Q  t )  +Q  q ) )
6463breq1d 3997 . . . . . . . . . . . 12  |-  ( r  =  ( s  +Q  t )  ->  (
( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( (
s  +Q  t )  +Q  q )  <Q 
( ( F `  q )  +Q  S
) ) )
6564ad3antlr 490 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  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 166 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( s  +Q  q
)  <Q  ( F `  q ) )  -> 
( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) )
6766ex 114 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  ->  ( ( s  +Q  q )  <Q  ( F `  q )  ->  ( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) ) )
6867reximdva 2572 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( E. q  e. 
Q.  ( s  +Q  q )  <Q  ( F `  q )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) ) )
6946, 68mpd 13 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) )
70 oveq1 5857 . . . . . . . . . 10  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
7170breq1d 3997 . . . . . . . . 9  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( r  +Q  q )  <Q  (
( F `  q
)  +Q  S ) ) )
7271rexbidv 2471 . . . . . . . 8  |-  ( l  =  r  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  E. q  e.  Q.  ( r  +Q  q )  <Q  (
( F `  q
)  +Q  S ) ) )
7318rabex 4131 . . . . . . . . 9  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  _V
7418rabex 4131 . . . . . . . . 9  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  _V
7573, 74op1st 6122 . . . . . . . 8  |-  ( 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 } >. )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) }
7672, 75elrab2 2889 . . . . . . 7  |-  ( r  e.  ( 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 } >. )  <-> 
( r  e.  Q.  /\ 
E. q  e.  Q.  ( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S ) ) )
7745, 69, 76sylanbrc 415 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  ( 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 } >. ) )
7877ex 114 . . . . 5  |-  ( ( ( ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 1st `  L )  /\  t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( r  =  ( s  +Q  t )  ->  r  e.  ( 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 } >. ) ) )
7978rexlimdvva 2595 . . . 4  |-  ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  ( E. s  e.  ( 1st `  L ) E. t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t )  -> 
r  e.  ( 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 } >. ) ) )
8013, 79mpd 13 . . 3  |-  ( (
ph  /\  r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  r  e.  ( 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 } >. ) )
8180ex 114 . 2  |-  ( ph  ->  ( r  e.  ( 1st `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  ->  r  e.  ( 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 } >. ) ) )
8281ssrdv 3153 1  |-  ( 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 } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 973    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449   {crab 2452    C_ wss 3121   <.cop 3584   class class class wbr 3987   -->wf 5192   ` cfv 5196  (class class class)co 5850   1stc1st 6114   Q.cnq 7229    +Q cplq 7231    <Q cltq 7234   P.cnp 7240    +P. cpp 7242
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-eprel 4272  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-irdg 6346  df-1o 6392  df-oadd 6396  df-omul 6397  df-er 6509  df-ec 6511  df-qs 6515  df-ni 7253  df-pli 7254  df-mi 7255  df-lti 7256  df-plpq 7293  df-mpq 7294  df-enq 7296  df-nqqs 7297  df-plqqs 7298  df-mqqs 7299  df-1nqqs 7300  df-rq 7301  df-ltnqqs 7302  df-inp 7415  df-iplp 7417
This theorem is referenced by:  cauappcvgprlemladdru  7605  cauappcvgprlemladd  7607
  Copyright terms: Public domain W3C validator