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

Theorem cauappcvgprlemopl 6744
Description: Lemma for cauappcvgpr 6760. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-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 } >.
Assertion
Ref Expression
cauappcvgprlemopl  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. r  e.  Q.  ( s  <Q 
r  /\  r  e.  ( 1st `  L ) ) )
Distinct variable groups:    A, p    L, p, q    ph, p, q    L, r, s    A, s, p    F, l, u, p, q, r, s    ph, r,
s
Allowed substitution hints:    ph( u, l)    A( u, r, q, l)    L( u, l)

Proof of Theorem cauappcvgprlemopl
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 oveq1 5519 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
21breq1d 3774 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
32rexbidv 2327 . . . . 5  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
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 } >.
54fveq2i 5181 . . . . . 6  |-  ( 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 } >. )
6 nqex 6461 . . . . . . . 8  |-  Q.  e.  _V
76rabex 3901 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
86rabex 3901 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
97, 8op1st 5773 . . . . . 6  |-  ( 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 ) }
105, 9eqtri 2060 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
113, 10elrab2 2700 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
1211simprbi 260 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
1312adantl 262 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
14 simprr 484 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  ->  ( s  +Q  q )  <Q  ( F `  q )
)
15 ltbtwnnqq 6513 . . . 4  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  <->  E. t  e.  Q.  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) )
1614, 15sylib 127 . . 3  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  ->  E. t  e.  Q.  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) )
17 simplrl 487 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  q  e.  Q. )
1811simplbi 259 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1918ad3antlr 462 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  s  e.  Q. )
20 ltaddnq 6505 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  q  <Q  ( q  +Q  s ) )
2117, 19, 20syl2anc 391 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  q  <Q  (
q  +Q  s ) )
22 addcomnqg 6479 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  ( q  +Q  s
)  =  ( s  +Q  q ) )
2317, 19, 22syl2anc 391 . . . . . . 7  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  ( q  +Q  s )  =  ( s  +Q  q ) )
2421, 23breqtrd 3788 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  q  <Q  (
s  +Q  q ) )
25 simprrl 491 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  ( s  +Q  q )  <Q  t
)
26 ltsonq 6496 . . . . . . 7  |-  <Q  Or  Q.
27 ltrelnq 6463 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2826, 27sotri 4720 . . . . . 6  |-  ( ( q  <Q  ( s  +Q  q )  /\  (
s  +Q  q ) 
<Q  t )  ->  q  <Q  t )
2924, 25, 28syl2anc 391 . . . . 5  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  q  <Q  t
)
30 simprl 483 . . . . . 6  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  t  e.  Q. )
31 ltexnqq 6506 . . . . . 6  |-  ( ( q  e.  Q.  /\  t  e.  Q. )  ->  ( q  <Q  t  <->  E. r  e.  Q.  (
q  +Q  r )  =  t ) )
3217, 30, 31syl2anc 391 . . . . 5  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  ( q  <Q 
t  <->  E. r  e.  Q.  ( q  +Q  r
)  =  t ) )
3329, 32mpbid 135 . . . 4  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  E. r  e.  Q.  ( q  +Q  r
)  =  t )
3425ad2antrr 457 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( s  +Q  q
)  <Q  t )
3519ad2antrr 457 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
s  e.  Q. )
3617ad2antrr 457 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
q  e.  Q. )
37 addcomnqg 6479 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  q  e.  Q. )  ->  ( s  +Q  q
)  =  ( q  +Q  s ) )
3835, 36, 37syl2anc 391 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( s  +Q  q
)  =  ( q  +Q  s ) )
3938breq1d 3774 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( ( s  +Q  q )  <Q  t  <->  ( q  +Q  s ) 
<Q  t ) )
4034, 39mpbid 135 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( q  +Q  s
)  <Q  t )
41 simpr 103 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( q  +Q  r
)  =  t )
4240, 41breqtrrd 3790 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( q  +Q  s
)  <Q  ( q  +Q  r ) )
43 simplr 482 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
r  e.  Q. )
44 ltanqg 6498 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  r  e.  Q.  /\  q  e.  Q. )  ->  (
s  <Q  r  <->  ( q  +Q  s )  <Q  (
q  +Q  r ) ) )
4535, 43, 36, 44syl3anc 1135 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( s  <Q  r  <->  ( q  +Q  s ) 
<Q  ( q  +Q  r
) ) )
4642, 45mpbird 156 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
s  <Q  r )
47 simprrr 492 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  t  <Q  ( F `  q )
)
4847ad2antrr 457 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
t  <Q  ( F `  q ) )
49 addcomnqg 6479 . . . . . . . . . . . . 13  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  +Q  r
)  =  ( r  +Q  q ) )
5036, 43, 49syl2anc 391 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( q  +Q  r
)  =  ( r  +Q  q ) )
5150, 41eqtr3d 2074 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( r  +Q  q
)  =  t )
5251breq1d 3774 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( ( r  +Q  q )  <Q  ( F `  q )  <->  t 
<Q  ( F `  q
) ) )
5348, 52mpbird 156 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( r  +Q  q
)  <Q  ( F `  q ) )
54 rspe 2370 . . . . . . . . 9  |-  ( ( q  e.  Q.  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( F `  q ) )
5536, 53, 54syl2anc 391 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( F `  q ) )
56 oveq1 5519 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
5756breq1d 3774 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
5857rexbidv 2327 . . . . . . . . 9  |-  ( l  =  r  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
5958, 10elrab2 2700 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
6043, 55, 59sylanbrc 394 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
r  e.  ( 1st `  L ) )
6146, 60jca 290 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  (
s  +Q  q ) 
<Q  ( F `  q
) ) )  /\  ( t  e.  Q.  /\  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) ) )  /\  r  e.  Q. )  /\  (
q  +Q  r )  =  t )  -> 
( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
6261ex 108 . . . . 5  |-  ( ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  /\  r  e.  Q. )  ->  ( ( q  +Q  r )  =  t  ->  ( s  <Q  r  /\  r  e.  ( 1st `  L
) ) ) )
6362reximdva 2421 . . . 4  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  ( E. r  e.  Q.  ( q  +Q  r )  =  t  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) ) )
6433, 63mpd 13 . . 3  |-  ( ( ( ( ph  /\  s  e.  ( 1st `  L ) )  /\  ( q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  /\  ( t  e. 
Q.  /\  ( (
s  +Q  q ) 
<Q  t  /\  t  <Q  ( F `  q
) ) ) )  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
6516, 64rexlimddv 2437 . 2  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  ->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )
6613, 65rexlimddv 2437 1  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. r  e.  Q.  ( s  <Q 
r  /\  r  e.  ( 1st `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   Q.cnq 6378    +Q cplq 6380    <Q cltq 6383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451
This theorem is referenced by:  cauappcvgprlemrnd  6748
  Copyright terms: Public domain W3C validator