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Theorem cauappcvgprlemladdfl 7667
Description: Lemma for cauappcvgprlemladd 7670. 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 7665 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 cauappcvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 7559 . . . . . . 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 7480 . . . . . . 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 7387 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvl 7524 . . . . . 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 411 . . . . 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 296 . . . 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 5895 . . . . . . . . . . . . . . . 16  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
1514breq1d 4025 . . . . . . . . . . . . . . 15  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
1615rexbidv 2488 . . . . . . . . . . . . . 14  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
174fveq2i 5530 . . . . . . . . . . . . . . 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 7375 . . . . . . . . . . . . . . . . 17  |-  Q.  e.  _V
1918rabex 4159 . . . . . . . . . . . . . . . 16  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2018rabex 4159 . . . . . . . . . . . . . . . 16  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2119, 20op1st 6160 . . . . . . . . . . . . . . 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 2208 . . . . . . . . . . . . . 14  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
2316, 22elrab2 2908 . . . . . . . . . . . . 13  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
2423biimpi 120 . . . . . . . . . . . 12  |-  ( s  e.  ( 1st `  L
)  ->  ( s  e.  Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
2524ad2antrl 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 } >. ) ) )  -> 
( s  e.  Q.  /\ 
E. q  e.  Q.  ( s  +Q  q
)  <Q  ( F `  q ) ) )
2625adantr 276 . . . . . . . . . 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 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 ) )  -> 
s  e.  Q. )
28 vex 2752 . . . . . . . . . . . . . . 15  |-  t  e. 
_V
29 breq1 4018 . . . . . . . . . . . . . . 15  |-  ( l  =  t  ->  (
l  <Q  S  <->  t  <Q  S ) )
30 ltnqex 7561 . . . . . . . . . . . . . . . 16  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 7562 . . . . . . . . . . . . . . . 16  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op1st 6160 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { l  |  l 
<Q  S }
3328, 29, 32elab2 2897 . . . . . . . . . . . . . 14  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  t  <Q  S )
3433biimpi 120 . . . . . . . . . . . . 13  |-  ( t  e.  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  t  <Q  S )
3534ad2antll 491 . . . . . . . . . . . 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 276 . . . . . . . . . . 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 7377 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
3837brel 4690 . . . . . . . . . . 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 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 ) )  -> 
t  e.  Q. )
41 addclnq 7387 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4227, 40, 41syl2anc 411 . . . . . . . 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 2250 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4443adantl 277 . . . . . . . 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 167 . . . . . . 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 114 . . . . . . . 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 488 . . . . . . . . . . . . 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 528 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 7393 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
5150adantl 277 . . . . . . . . . . . . 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 7394 . . . . . . . . . . . . . 14  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f  +Q  g
)  +Q  h )  =  ( f  +Q  ( g  +Q  h
) ) )
5352adantl 277 . . . . . . . . . . . . 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 6068 . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 4690 . . . . . . . . . . . . . . 15  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  ->  ( (
s  +Q  q )  e.  Q.  /\  ( F `  q )  e.  Q. ) )
58 lt2addnq 7416 . . . . . . . . . . . . . . 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 290 . . . . . . . . . . . . . 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 433 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 4039 . . . . . . . . . . 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 5895 . . . . . . . . . . . . 13  |-  ( r  =  ( s  +Q  t )  ->  (
r  +Q  q )  =  ( ( s  +Q  t )  +Q  q ) )
6463breq1d 4025 . . . . . . . . . . . 12  |-  ( r  =  ( s  +Q  t )  ->  (
( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( (
s  +Q  t )  +Q  q )  <Q 
( ( F `  q )  +Q  S
) ) )
6564ad3antlr 493 . . . . . . . . . . 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 167 . . . . . . . . . 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 115 . . . . . . . . 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 2589 . . . . . . . 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 5895 . . . . . . . . . 10  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
7170breq1d 4025 . . . . . . . . 9  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( r  +Q  q )  <Q  (
( F `  q
)  +Q  S ) ) )
7271rexbidv 2488 . . . . . . . 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 4159 . . . . . . . . 9  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  _V
7418rabex 4159 . . . . . . . . 9  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  _V
7573, 74op1st 6160 . . . . . . . 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 2908 . . . . . . 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 417 . . . . . 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 115 . . . . 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 2612 . . . 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 115 . 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 3173 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 104    <-> wb 105    /\ w3a 979    = wceq 1363    e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   {crab 2469    C_ wss 3141   <.cop 3607   class class class wbr 4015   -->wf 5224   ` cfv 5228  (class class class)co 5888   1stc1st 6152   Q.cnq 7292    +Q cplq 7294    <Q cltq 7297   P.cnp 7303    +P. cpp 7305
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-iplp 7480
This theorem is referenced by:  cauappcvgprlemladdru  7668  cauappcvgprlemladd  7670
  Copyright terms: Public domain W3C validator