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Theorem cauappcvgprlemladdfl 7475
Description: Lemma for cauappcvgprlemladd 7478. 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 7473 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 cauappcvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 7367 . . . . . . 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 7288 . . . . . . 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 7195 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvl 7332 . . . . . 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 408 . . . . 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 5781 . . . . . . . . . . . . . . . 16  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
1514breq1d 3939 . . . . . . . . . . . . . . 15  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
1615rexbidv 2438 . . . . . . . . . . . . . 14  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
174fveq2i 5424 . . . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . . . . . 17  |-  Q.  e.  _V
1918rabex 4072 . . . . . . . . . . . . . . . 16  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
2018rabex 4072 . . . . . . . . . . . . . . . 16  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2119, 20op1st 6044 . . . . . . . . . . . . . . 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 2160 . . . . . . . . . . . . . 14  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
2316, 22elrab2 2843 . . . . . . . . . . . . 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 481 . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . 15  |-  t  e. 
_V
29 breq1 3932 . . . . . . . . . . . . . . 15  |-  ( l  =  t  ->  (
l  <Q  S  <->  t  <Q  S ) )
30 ltnqex 7369 . . . . . . . . . . . . . . . 16  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 7370 . . . . . . . . . . . . . . . 16  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op1st 6044 . . . . . . . . . . . . . . 15  |-  ( 1st `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { l  |  l 
<Q  S }
3328, 29, 32elab2 2832 . . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 7185 . . . . . . . . . . . 12  |-  <Q  C_  ( Q.  X.  Q. )
3837brel 4591 . . . . . . . . . . 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 7195 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4227, 40, 41syl2anc 408 . . . . . . . 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 2202 . . . . . . . . 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 479 . . . . . . . . . . . . 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 519 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 7201 . . . . . . . . . . . . . 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 7202 . . . . . . . . . . . . . 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 5951 . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 4591 . . . . . . . . . . . . . . 15  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  ->  ( (
s  +Q  q )  e.  Q.  /\  ( F `  q )  e.  Q. ) )
58 lt2addnq 7224 . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . 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 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 } >. ) ) )  /\  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 3952 . . . . . . . . . . 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 5781 . . . . . . . . . . . . 13  |-  ( r  =  ( s  +Q  t )  ->  (
r  +Q  q )  =  ( ( s  +Q  t )  +Q  q ) )
6463breq1d 3939 . . . . . . . . . . . 12  |-  ( r  =  ( s  +Q  t )  ->  (
( r  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( (
s  +Q  t )  +Q  q )  <Q 
( ( F `  q )  +Q  S
) ) )
6564ad3antlr 484 . . . . . . . . . . 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 2534 . . . . . . . 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 5781 . . . . . . . . . 10  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
7170breq1d 3939 . . . . . . . . 9  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( ( F `
 q )  +Q  S )  <->  ( r  +Q  q )  <Q  (
( F `  q
)  +Q  S ) ) )
7271rexbidv 2438 . . . . . . . 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 4072 . . . . . . . . 9  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  _V
7418rabex 4072 . . . . . . . . 9  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  _V
7573, 74op1st 6044 . . . . . . . 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 2843 . . . . . . 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 413 . . . . . 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 2557 . . . 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 3103 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 962    = wceq 1331    e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   {crab 2420    C_ wss 3071   <.cop 3530   class class class wbr 3929   -->wf 5119   ` cfv 5123  (class class class)co 5774   1stc1st 6036   Q.cnq 7100    +Q cplq 7102    <Q cltq 7105   P.cnp 7111    +P. cpp 7113
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-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-inp 7286  df-iplp 7288
This theorem is referenced by:  cauappcvgprlemladdru  7476  cauappcvgprlemladd  7478
  Copyright terms: Public domain W3C validator