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Theorem cauappcvgprlemladdfl 6811
Description: Lemma for cauappcvgprlemladd 6814. 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 6809 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 cauappcvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 6703 . . . . . . 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 6624 . . . . . . 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 6531 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvl 6668 . . . . . 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 397 . . . . 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 284 . . . 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 5547 . . . . . . . . . . . . . . . 16  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
1514breq1d 3802 . . . . . . . . . . . . . . 15  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
1615rexbidv 2344 . . . . . . . . . . . . . 14  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
174fveq2i 5209 . . . . . . . . . . . . . . 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 6519 . . . . . . . . . . . . . . . . 17  |-  Q.  e.  _V
1918rabex 3929 . . . . . . . . . . . . . . . 16  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2018rabex 3929 . . . . . . . . . . . . . . . 16  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2119, 20op1st 5801 . . . . . . . . . . . . . . 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 2076 . . . . . . . . . . . . . 14  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
2316, 22elrab2 2723 . . . . . . . . . . . . 13  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
2423biimpi 117 . . . . . . . . . . . 12  |-  ( s  e.  ( 1st `  L
)  ->  ( s  e.  Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
2524ad2antrl 467 . . . . . . . . . . 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 265 . . . . . . . . . 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 109 . . . . . . . . 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 2577 . . . . . . . . . . . . . . 15  |-  t  e. 
_V
29 breq1 3795 . . . . . . . . . . . . . . 15  |-  ( l  =  t  ->  (
l  <Q  S  <->  t  <Q  S ) )
30 ltnqex 6705 . . . . . . . . . . . . . . . 16  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 6706 . . . . . . . . . . . . . . . 16  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op1st 5801 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { l  |  l 
<Q  S }
3328, 29, 32elab2 2713 . . . . . . . . . . . . . 14  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  t  <Q  S )
3433biimpi 117 . . . . . . . . . . . . 13  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  t  <Q  S )
3534ad2antll 468 . . . . . . . . . . . 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 265 . . . . . . . . . . 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 6521 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
3837brel 4420 . . . . . . . . . . 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 109 . . . . . . . . 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 6531 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4227, 40, 41syl2anc 397 . . . . . . . 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 2116 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4443adantl 266 . . . . . . . 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 160 . . . . . . 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 111 . . . . . . . 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 465 . . . . . . . . . . . . 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 490 . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 6537 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5150adantl 266 . . . . . . . . . . . . 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 6538 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5352adantl 266 . . . . . . . . . . . . 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 5709 . . . . . . . . . . . 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 107 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 4420 . . . . . . . . . . . . . . 15  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  ->  ( (
s  +Q  q )  e.  Q.  /\  ( F `  q )  e.  Q. ) )
58 lt2addnq 6560 . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . 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 417 . . . . . . . . . . . . 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 454 . . . . . . . . . . . 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 3814 . . . . . . . . . . 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 5547 . . . . . . . . . . . . 13  |-  ( r  =  ( s  +Q  t )  ->  (
r  +Q  q )  =  ( ( s  +Q  t )  +Q  q ) )
6463breq1d 3802 . . . . . . . . . . . 12  |-  ( r  =  ( s  +Q  t )  ->  (
( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( (
s  +Q  t )  +Q  q )  <Q 
( ( F `  q )  +Q  S
) ) )
6564ad3antlr 470 . . . . . . . . . . 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 160 . . . . . . . . . 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 112 . . . . . . . . 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 2438 . . . . . . . 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 5547 . . . . . . . . . 10  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
7170breq1d 3802 . . . . . . . . 9  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( r  +Q  q )  <Q  (
( F `  q
)  +Q  S ) ) )
7271rexbidv 2344 . . . . . . . 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 3929 . . . . . . . . 9  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  _V
7418rabex 3929 . . . . . . . . 9  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  _V
7573, 74op1st 5801 . . . . . . . 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 2723 . . . . . . 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 402 . . . . . 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 112 . . . . 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 2457 . . . 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 112 . 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 2979 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   {cab 2042   A.wral 2323   E.wrex 2324   {crab 2327    C_ wss 2945   <.cop 3406   class class class wbr 3792   -->wf 4926   ` cfv 4930  (class class class)co 5540   1stc1st 5793   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    +P. cpp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-iplp 6624
This theorem is referenced by:  cauappcvgprlemladdru  6812  cauappcvgprlemladd  6814
  Copyright terms: Public domain W3C validator