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Theorem cauappcvgprlemopl 6887
Description: Lemma for cauappcvgpr 6903. 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 5544 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
21breq1d 3797 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
32rexbidv 2370 . . . . 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 5206 . . . . . 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 6604 . . . . . . . 8  |-  Q.  e.  _V
76rabex 3924 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
86rabex 3924 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
97, 8op1st 5798 . . . . . 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 2102 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
113, 10elrab2 2752 . . . 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 6656 . . . 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 6648 . . . . . . . 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 6622 . . . . . . . 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 3811 . . . . . 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 6639 . . . . . . 7  |-  <Q  Or  Q.
27 ltrelnq 6606 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2826, 27sotri 4744 . . . . . 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 6649 . . . . . 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 6622 . . . . . . . . . . . 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 3797 . . . . . . . . . 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 3813 . . . . . . . 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 6641 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  r  e.  Q.  /\  q  e.  Q. )  ->  (
s  <Q  r  <->  ( q  +Q  s )  <Q  (
q  +Q  r ) ) )
4535, 43, 36, 44syl3anc 1170 . . . . . . . 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 6622 . . . . . . . . . . . . 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 2116 . . . . . . . . . . 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 3797 . . . . . . . . . 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 2413 . . . . . . . . 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 5544 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
5756breq1d 3797 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
5857rexbidv 2370 . . . . . . . . 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 2752 . . . . . . . 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 2464 . . . 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 2482 . 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 2482 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 1285    e. wcel 1434   A.wral 2349   E.wrex 2350   {crab 2353   <.cop 3403   class class class wbr 3787   -->wf 4922   ` cfv 4926  (class class class)co 5537   1stc1st 5790   Q.cnq 6521    +Q cplq 6523    <Q cltq 6526
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594
This theorem is referenced by:  cauappcvgprlemrnd  6891
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