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Theorem cauappcvgprlemopl 7184
Description: Lemma for cauappcvgpr 7200. 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 5641 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
21breq1d 3847 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
32rexbidv 2381 . . . . 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 5292 . . . . . 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 6901 . . . . . . . 8  |-  Q.  e.  _V
76rabex 3975 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
86rabex 3975 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
97, 8op1st 5899 . . . . . 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 2108 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
113, 10elrab2 2772 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
1211simprbi 269 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
1312adantl 271 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
14 simprr 499 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  ->  ( s  +Q  q )  <Q  ( F `  q )
)
15 ltbtwnnqq 6953 . . . 4  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  <->  E. t  e.  Q.  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) )
1614, 15sylib 120 . . 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 502 . . . . . . . 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 268 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1918ad3antlr 477 . . . . . . . 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 6945 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  q  <Q  ( q  +Q  s ) )
2117, 19, 20syl2anc 403 . . . . . . 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 6919 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  ( q  +Q  s
)  =  ( s  +Q  q ) )
2317, 19, 22syl2anc 403 . . . . . . 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 3861 . . . . . 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 506 . . . . . 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 6936 . . . . . . 7  |-  <Q  Or  Q.
27 ltrelnq 6903 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2826, 27sotri 4814 . . . . . 6  |-  ( ( q  <Q  ( s  +Q  q )  /\  (
s  +Q  q ) 
<Q  t )  ->  q  <Q  t )
2924, 25, 28syl2anc 403 . . . . 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 498 . . . . . 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 6946 . . . . . 6  |-  ( ( q  e.  Q.  /\  t  e.  Q. )  ->  ( q  <Q  t  <->  E. r  e.  Q.  (
q  +Q  r )  =  t ) )
3217, 30, 31syl2anc 403 . . . . 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 145 . . . 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 472 . . . . . . . . . 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 472 . . . . . . . . . . . 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 472 . . . . . . . . . . . 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 6919 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  q  e.  Q. )  ->  ( s  +Q  q
)  =  ( q  +Q  s ) )
3835, 36, 37syl2anc 403 . . . . . . . . . . 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 3847 . . . . . . . . . 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 145 . . . . . . . . 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 108 . . . . . . . . 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 3863 . . . . . . . 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 497 . . . . . . . . 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 6938 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  r  e.  Q.  /\  q  e.  Q. )  ->  (
s  <Q  r  <->  ( q  +Q  s )  <Q  (
q  +Q  r ) ) )
4535, 43, 36, 44syl3anc 1174 . . . . . . . 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 165 . . . . . . 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 507 . . . . . . . . . . 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 472 . . . . . . . . . 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 6919 . . . . . . . . . . . . 13  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  +Q  r
)  =  ( r  +Q  q ) )
5036, 43, 49syl2anc 403 . . . . . . . . . . . 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 2122 . . . . . . . . . . 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 3847 . . . . . . . . . 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 165 . . . . . . . . 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 2424 . . . . . . . . 9  |-  ( ( q  e.  Q.  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( F `  q ) )
5536, 53, 54syl2anc 403 . . . . . . . 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 5641 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
5756breq1d 3847 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
5857rexbidv 2381 . . . . . . . . 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 2772 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
6043, 55, 59sylanbrc 408 . . . . . . 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 300 . . . . . 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 113 . . . . 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 2475 . . . 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 2493 . 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 2493 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3444   class class class wbr 3837   -->wf 4998   ` cfv 5002  (class class class)co 5634   1stc1st 5891   Q.cnq 6818    +Q cplq 6820    <Q cltq 6823
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
This theorem is referenced by:  cauappcvgprlemrnd  7188
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