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Theorem cauappcvgprlemopl 7474
Description: Lemma for cauappcvgpr 7490. 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 5785 . . . . . . 7  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
21breq1d 3943 . . . . . 6  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
32rexbidv 2439 . . . . 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 5428 . . . . . 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 7191 . . . . . . . 8  |-  Q.  e.  _V
76rabex 4076 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
86rabex 4076 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
97, 8op1st 6048 . . . . . 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 2161 . . . . 5  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
113, 10elrab2 2844 . . . 4  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
1211simprbi 273 . . 3  |-  ( s  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
1312adantl 275 . 2  |-  ( (
ph  /\  s  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
14 simprr 522 . . . 4  |-  ( ( ( ph  /\  s  e.  ( 1st `  L
) )  /\  (
q  e.  Q.  /\  ( s  +Q  q
)  <Q  ( F `  q ) ) )  ->  ( s  +Q  q )  <Q  ( F `  q )
)
15 ltbtwnnqq 7243 . . . 4  |-  ( ( s  +Q  q ) 
<Q  ( F `  q
)  <->  E. t  e.  Q.  ( ( s  +Q  q )  <Q  t  /\  t  <Q  ( F `
 q ) ) )
1614, 15sylib 121 . . 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 525 . . . . . . . 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 272 . . . . . . . . 9  |-  ( s  e.  ( 1st `  L
)  ->  s  e.  Q. )
1918ad3antlr 485 . . . . . . . 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 7235 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  q  <Q  ( q  +Q  s ) )
2117, 19, 20syl2anc 409 . . . . . . 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 7209 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  s  e.  Q. )  ->  ( q  +Q  s
)  =  ( s  +Q  q ) )
2317, 19, 22syl2anc 409 . . . . . . 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 3958 . . . . . 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 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
) ) ) )  ->  ( s  +Q  q )  <Q  t
)
26 ltsonq 7226 . . . . . . 7  |-  <Q  Or  Q.
27 ltrelnq 7193 . . . . . . 7  |-  <Q  C_  ( Q.  X.  Q. )
2826, 27sotri 4938 . . . . . 6  |-  ( ( q  <Q  ( s  +Q  q )  /\  (
s  +Q  q ) 
<Q  t )  ->  q  <Q  t )
2924, 25, 28syl2anc 409 . . . . 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 521 . . . . . 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 7236 . . . . . 6  |-  ( ( q  e.  Q.  /\  t  e.  Q. )  ->  ( q  <Q  t  <->  E. r  e.  Q.  (
q  +Q  r )  =  t ) )
3217, 30, 31syl2anc 409 . . . . 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 146 . . . 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 480 . . . . . . . . . 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 480 . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 7209 . . . . . . . . . . . 12  |-  ( ( s  e.  Q.  /\  q  e.  Q. )  ->  ( s  +Q  q
)  =  ( q  +Q  s ) )
3835, 36, 37syl2anc 409 . . . . . . . . . . 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 3943 . . . . . . . . . 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 146 . . . . . . . . 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 109 . . . . . . . . 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 3960 . . . . . . . 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 520 . . . . . . . . 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 7228 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  r  e.  Q.  /\  q  e.  Q. )  ->  (
s  <Q  r  <->  ( q  +Q  s )  <Q  (
q  +Q  r ) ) )
4535, 43, 36, 44syl3anc 1217 . . . . . . . 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 166 . . . . . . 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 530 . . . . . . . . . . 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 480 . . . . . . . . . 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 7209 . . . . . . . . . . . . 13  |-  ( ( q  e.  Q.  /\  r  e.  Q. )  ->  ( q  +Q  r
)  =  ( r  +Q  q ) )
5036, 43, 49syl2anc 409 . . . . . . . . . . . 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 2175 . . . . . . . . . . 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 3943 . . . . . . . . . 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 166 . . . . . . . . 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 2482 . . . . . . . . 9  |-  ( ( q  e.  Q.  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  ->  E. q  e.  Q.  ( r  +Q  q
)  <Q  ( F `  q ) )
5536, 53, 54syl2anc 409 . . . . . . . 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 5785 . . . . . . . . . . 11  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
5756breq1d 3943 . . . . . . . . . 10  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
5857rexbidv 2439 . . . . . . . . 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 2844 . . . . . . . 8  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
6043, 55, 59sylanbrc 414 . . . . . . 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 304 . . . . . 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 114 . . . . 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 2535 . . . 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 2555 . 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 2555 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 103    <-> wb 104    = wceq 1332    e. wcel 1481   A.wral 2417   E.wrex 2418   {crab 2421   <.cop 3531   class class class wbr 3933   -->wf 5123   ` cfv 5127  (class class class)co 5778   1stc1st 6040   Q.cnq 7108    +Q cplq 7110    <Q cltq 7113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-iinf 4506
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-tr 4031  df-eprel 4215  df-id 4219  df-po 4222  df-iso 4223  df-iord 4292  df-on 4294  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-recs 6206  df-irdg 6271  df-1o 6317  df-oadd 6321  df-omul 6322  df-er 6433  df-ec 6435  df-qs 6439  df-ni 7132  df-pli 7133  df-mi 7134  df-lti 7135  df-plpq 7172  df-mpq 7173  df-enq 7175  df-nqqs 7176  df-plqqs 7177  df-mqqs 7178  df-1nqqs 7179  df-rq 7180  df-ltnqqs 7181
This theorem is referenced by:  cauappcvgprlemrnd  7478
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