ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cauappcvgprlemladdfu Unicode version

Theorem cauappcvgprlemladdfu 7192
Description: Lemma for cauappcvgprlemladd 7196. The forward subset relationship for the upper 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
cauappcvgprlemladdfu  |-  ( 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 } >. ) )
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 cauappcvgprlemladdfu
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 7191 . . . . . 6  |-  ( ph  ->  L  e.  P. )
6 cauappcvgprlemladd.s . . . . . . 7  |-  ( ph  ->  S  e.  Q. )
7 nqprlu 7085 . . . . . . 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 7006 . . . . . . 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 6913 . . . . . . 7  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
119, 10genpelvu 7051 . . . . . 6  |-  ( ( L  e.  P.  /\  <. { l  |  l 
<Q  S } ,  {
u  |  S  <Q  u } >.  e.  P. )  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
125, 8, 11syl2anc 403 . . . . 5  |-  ( ph  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  <->  E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t ) ) )
1312biimpa 290 . . . 4  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  E. s  e.  ( 2nd `  L
) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) r  =  ( s  +Q  t ) )
14 breq2 3841 . . . . . . . . . . . . . . . 16  |-  ( u  =  s  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  s
) )
1514rexbidv 2381 . . . . . . . . . . . . . . 15  |-  ( u  =  s  ->  ( E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
164fveq2i 5292 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  L )  =  ( 2nd `  <. { 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 } >. )
17 nqex 6901 . . . . . . . . . . . . . . . . . 18  |-  Q.  e.  _V
1817rabex 3975 . . . . . . . . . . . . . . . . 17  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
1917rabex 3975 . . . . . . . . . . . . . . . . 17  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
2018, 19op2nd 5900 . . . . . . . . . . . . . . . 16  |-  ( 2nd `  <. { 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 } >. )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2116, 20eqtri 2108 . . . . . . . . . . . . . . 15  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
2215, 21elrab2 2772 . . . . . . . . . . . . . 14  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
2322biimpi 118 . . . . . . . . . . . . 13  |-  ( s  e.  ( 2nd `  L
)  ->  ( s  e.  Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
2423adantr 270 . . . . . . . . . . . 12  |-  ( ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) )  ->  (
s  e.  Q.  /\  E. q  e.  Q.  (
( F `  q
)  +Q  q ) 
<Q  s ) )
2524adantl 271 . . . . . . . . . . 11  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( s  e.  Q.  /\ 
E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  s ) )
2625adantr 270 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  e.  Q.  /\ 
E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  s ) )
2726simpld 110 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
s  e.  Q. )
28 vex 2622 . . . . . . . . . . . . . 14  |-  t  e. 
_V
29 breq2 3841 . . . . . . . . . . . . . 14  |-  ( u  =  t  ->  ( S  <Q  u  <->  S  <Q  t ) )
30 ltnqex 7087 . . . . . . . . . . . . . . 15  |-  { l  |  l  <Q  S }  e.  _V
31 gtnqex 7088 . . . . . . . . . . . . . . 15  |-  { u  |  S  <Q  u }  e.  _V
3230, 31op2nd 5900 . . . . . . . . . . . . . 14  |-  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  =  { u  |  S  <Q  u }
3328, 29, 32elab2 2761 . . . . . . . . . . . . 13  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  <->  S  <Q  t )
34 ltrelnq 6903 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
3534brel 4478 . . . . . . . . . . . . 13  |-  ( S 
<Q  t  ->  ( S  e.  Q.  /\  t  e.  Q. ) )
3633, 35sylbi 119 . . . . . . . . . . . 12  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  ( S  e.  Q.  /\  t  e. 
Q. ) )
3736simprd 112 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  t  e.  Q. )
3837ad2antll 475 . . . . . . . . . 10  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
t  e.  Q. )
3938adantr 270 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
t  e.  Q. )
40 addclnq 6913 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  t  e.  Q. )  ->  ( s  +Q  t
)  e.  Q. )
4127, 39, 40syl2anc 403 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( s  +Q  t
)  e.  Q. )
42 eleq1 2150 . . . . . . . . 9  |-  ( r  =  ( s  +Q  t )  ->  (
r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4342adantl 271 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( r  e.  Q.  <->  ( s  +Q  t )  e.  Q. ) )
4441, 43mpbird 165 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  Q. )
4526simprd 112 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  s )
4633biimpi 118 . . . . . . . . . . . . . . . 16  |-  ( t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )  ->  S  <Q  t )
4746ad2antll 475 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  ->  S  <Q  t )
4847adantr 270 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  S  <Q  t )
4948ad2antrr 472 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  ->  S  <Q  t )
506ad5antr 480 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  ->  S  e.  Q. )
5139ad2antrr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
t  e.  Q. )
521ad5antr 480 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  ->  F : Q. --> Q. )
53 simplr 497 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
q  e.  Q. )
5452, 53ffvelrnd 5419 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( F `  q
)  e.  Q. )
55 addclnq 6913 . . . . . . . . . . . . . . 15  |-  ( ( ( F `  q
)  e.  Q.  /\  q  e.  Q. )  ->  ( ( F `  q )  +Q  q
)  e.  Q. )
5654, 53, 55syl2anc 403 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( F `  q )  +Q  q
)  e.  Q. )
57 ltanqg 6938 . . . . . . . . . . . . . 14  |-  ( ( S  e.  Q.  /\  t  e.  Q.  /\  (
( F `  q
)  +Q  q )  e.  Q. )  -> 
( S  <Q  t  <->  ( ( ( F `  q )  +Q  q
)  +Q  S ) 
<Q  ( ( ( F `
 q )  +Q  q )  +Q  t
) ) )
5850, 51, 56, 57syl3anc 1174 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( S  <Q  t  <->  ( ( ( F `  q )  +Q  q
)  +Q  S ) 
<Q  ( ( ( F `
 q )  +Q  q )  +Q  t
) ) )
5949, 58mpbid 145 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  ( ( ( F `  q )  +Q  q )  +Q  t ) )
60 simpr 108 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( F `  q )  +Q  q
)  <Q  s )
61 ltanqg 6938 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
6261adantl 271 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
6327ad2antrr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
s  e.  Q. )
64 addcomnqg 6919 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
6564adantl 271 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
6662, 56, 63, 51, 65caovord2d 5796 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( ( F `
 q )  +Q  q )  <Q  s  <->  ( ( ( F `  q )  +Q  q
)  +Q  t ) 
<Q  ( s  +Q  t
) ) )
6760, 66mpbid 145 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( ( F `
 q )  +Q  q )  +Q  t
)  <Q  ( s  +Q  t ) )
68 ltsonq 6936 . . . . . . . . . . . . 13  |-  <Q  Or  Q.
6968, 34sotri 4814 . . . . . . . . . . . 12  |-  ( ( ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  ( ( ( F `  q )  +Q  q )  +Q  t )  /\  (
( ( F `  q )  +Q  q
)  +Q  t ) 
<Q  ( s  +Q  t
) )  ->  (
( ( F `  q )  +Q  q
)  +Q  S ) 
<Q  ( s  +Q  t
) )
7059, 67, 69syl2anc 403 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  ( s  +Q  t ) )
71 simpllr 501 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
r  =  ( s  +Q  t ) )
7270, 71breqtrrd 3863 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  /\  ( ( F `  q )  +Q  q
)  <Q  s )  -> 
( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  r )
7372ex 113 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  /\  q  e.  Q. )  ->  ( ( ( F `
 q )  +Q  q )  <Q  s  ->  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  r ) )
7473reximdva 2475 . . . . . . . 8  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
( E. q  e. 
Q.  ( ( F `
 q )  +Q  q )  <Q  s  ->  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  r ) )
7545, 74mpd 13 . . . . . . 7  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  ->  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  r )
76 breq2 3841 . . . . . . . . 9  |-  ( u  =  r  ->  (
( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u  <->  ( (
( F `  q
)  +Q  q )  +Q  S )  <Q 
r ) )
7776rexbidv 2381 . . . . . . . 8  |-  ( u  =  r  ->  ( E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u  <->  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  r
) )
7817rabex 3975 . . . . . . . . 9  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( ( F `  q )  +Q  S
) }  e.  _V
7917rabex 3975 . . . . . . . . 9  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `  q )  +Q  q )  +Q  S )  <Q  u }  e.  _V
8078, 79op2nd 5900 . . . . . . . 8  |-  ( 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 } >. )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  u }
8177, 80elrab2 2772 . . . . . . 7  |-  ( r  e.  ( 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 } >. )  <-> 
( r  e.  Q.  /\ 
E. q  e.  Q.  ( ( ( F `
 q )  +Q  q )  +Q  S
)  <Q  r ) )
8244, 75, 81sylanbrc 408 . . . . . 6  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  /\  ( s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  r  =  ( s  +Q  t ) )  -> 
r  e.  ( 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 } >. ) )
8382ex 113 . . . . 5  |-  ( ( ( ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  /\  (
s  e.  ( 2nd `  L )  /\  t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. ) ) )  -> 
( r  =  ( s  +Q  t )  ->  r  e.  ( 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 } >. ) ) )
8483rexlimdvva 2496 . . . 4  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  ( E. s  e.  ( 2nd `  L ) E. t  e.  ( 2nd `  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
r  =  ( s  +Q  t )  -> 
r  e.  ( 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 } >. ) ) )
8513, 84mpd 13 . . 3  |-  ( (
ph  /\  r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
) )  ->  r  e.  ( 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 } >. ) )
8685ex 113 . 2  |-  ( ph  ->  ( r  e.  ( 2nd `  ( L  +P.  <. { l  |  l  <Q  S } ,  { u  |  S  <Q  u } >. )
)  ->  r  e.  ( 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 } >. ) ) )
8786ssrdv 3029 1  |-  ( 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 } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363    C_ wss 2997   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   2ndc2nd 5892   Q.cnq 6818    +Q cplq 6820    <Q cltq 6823   P.cnp 6829    +P. cpp 6831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-inp 7004  df-iplp 7006
This theorem is referenced by:  cauappcvgprlemladdrl  7195  cauappcvgprlemladd  7196
  Copyright terms: Public domain W3C validator