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Theorem cauappcvgprlemopl 7676
Description: Lemma for cauappcvgpr 7692. 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 5904 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
21breq1d 4028 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
32rexbidv 2491 . . . . 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 5537 . . . . . 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 7393 . . . . . . . 8  |-  Q.  e.  _V
76rabex 4162 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
86rabex 4162 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
97, 8op1st 6172 . . . . . 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 2210 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
113, 10elrab2 2911 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
1211simprbi 275 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
1312adantl 277 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
14 simprr 531 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  ->  ( s  +Q  q )  <Q  ( F `  q )
)
15 ltbtwnnqq 7445 . . . 4  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  <->  E. t  e.  Q.  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) )
1614, 15sylib 122 . . 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 535 . . . . . . . 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 274 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1918ad3antlr 493 . . . . . . . 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 7437 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  q  <Q  ( q  +Q  s ) )
2117, 19, 20syl2anc 411 . . . . . . 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 7411 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  ( q  +Q  s
)  =  ( s  +Q  q ) )
2317, 19, 22syl2anc 411 . . . . . . 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 4044 . . . . . 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 539 . . . . . 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 7428 . . . . . . 7  |-  <Q  Or  Q.
27 ltrelnq 7395 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2826, 27sotri 5042 . . . . . 6  |-  ( ( q  <Q  ( s  +Q  q )  /\  (
s  +Q  q ) 
<Q  t )  ->  q  <Q  t )
2924, 25, 28syl2anc 411 . . . . 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 529 . . . . . 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 7438 . . . . . 6  |-  ( ( q  e.  Q.  /\  t  e.  Q. )  ->  ( q  <Q  t  <->  E. r  e.  Q.  (
q  +Q  r )  =  t ) )
3217, 30, 31syl2anc 411 . . . . 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 147 . . . 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 488 . . . . . . . . . 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 488 . . . . . . . . . . . 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 488 . . . . . . . . . . . 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 7411 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  q  e.  Q. )  ->  ( s  +Q  q
)  =  ( q  +Q  s ) )
3835, 36, 37syl2anc 411 . . . . . . . . . . 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 4028 . . . . . . . . . 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 147 . . . . . . . . 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 110 . . . . . . . . 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 4046 . . . . . . . 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 528 . . . . . . . . 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 7430 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  r  e.  Q.  /\  q  e.  Q. )  ->  (
s  <Q  r  <->  ( q  +Q  s )  <Q  (
q  +Q  r ) ) )
4535, 43, 36, 44syl3anc 1249 . . . . . . . 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 167 . . . . . . 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 540 . . . . . . . . . . 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 488 . . . . . . . . . 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 7411 . . . . . . . . . . . . 13  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  +Q  r
)  =  ( r  +Q  q ) )
5036, 43, 49syl2anc 411 . . . . . . . . . . . 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 2224 . . . . . . . . . . 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 4028 . . . . . . . . . 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 167 . . . . . . . . 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 2539 . . . . . . . . 9  |-  ( ( q  e.  Q.  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( F `  q ) )
5536, 53, 54syl2anc 411 . . . . . . . 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 5904 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
5756breq1d 4028 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
5857rexbidv 2491 . . . . . . . . 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 2911 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
6043, 55, 59sylanbrc 417 . . . . . . 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 306 . . . . . 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 115 . . . . 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 2592 . . . 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 2612 . 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 2612 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 104    <-> wb 105    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   {crab 2472   <.cop 3610   class class class wbr 4018   -->wf 5231   ` cfv 5235  (class class class)co 5897   1stc1st 6164   Q.cnq 7310    +Q cplq 7312    <Q cltq 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383
This theorem is referenced by:  cauappcvgprlemrnd  7680
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