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Theorem cauappcvgprlemupu 7570
Description: Lemma for cauappcvgpr 7583. The upper cut of the putative limit is upper. (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
cauappcvgprlemupu  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  ( 2nd `  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 cauappcvgprlemupu
StepHypRef Expression
1 ltrelnq 7286 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4639 . . . 4  |-  ( s 
<Q  r  ->  ( s  e.  Q.  /\  r  e.  Q. ) )
32simprd 113 . . 3  |-  ( s 
<Q  r  ->  r  e. 
Q. )
433ad2ant2 1004 . 2  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  Q. )
5 breq2 3970 . . . . . . 7  |-  ( u  =  s  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  s
) )
65rexbidv 2458 . . . . . 6  |-  ( u  =  s  ->  ( E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
7 cauappcvgpr.lim . . . . . . . 8  |-  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 } >.
87fveq2i 5472 . . . . . . 7  |-  ( 2nd `  L )  =  ( 2nd `  <. { 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 } >. )
9 nqex 7284 . . . . . . . . 9  |-  Q.  e.  _V
109rabex 4109 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
119rabex 4109 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
1210, 11op2nd 6096 . . . . . . 7  |-  ( 2nd `  <. { 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 } >. )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
138, 12eqtri 2178 . . . . . 6  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
146, 13elrab2 2871 . . . . 5  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
1514simprbi 273 . . . 4  |-  ( s  e.  ( 2nd `  L
)  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
)
16153ad2ant3 1005 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
)
17 ltsonq 7319 . . . . . . 7  |-  <Q  Or  Q.
1817, 1sotri 4982 . . . . . 6  |-  ( ( ( ( F `  q )  +Q  q
)  <Q  s  /\  s  <Q  r )  ->  (
( F `  q
)  +Q  q ) 
<Q  r )
1918expcom 115 . . . . 5  |-  ( s 
<Q  r  ->  ( ( ( F `  q
)  +Q  q ) 
<Q  s  ->  ( ( F `  q )  +Q  q )  <Q 
r ) )
20193ad2ant2 1004 . . . 4  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  ( (
( F `  q
)  +Q  q ) 
<Q  s  ->  ( ( F `  q )  +Q  q )  <Q 
r ) )
2120reximdv 2558 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  ( E. q  e.  Q.  (
( F `  q
)  +Q  q ) 
<Q  s  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
) )
2216, 21mpd 13 . 2  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
)
23 breq2 3970 . . . 4  |-  ( u  =  r  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  r
) )
2423rexbidv 2458 . . 3  |-  ( u  =  r  ->  ( E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
) )
2524, 13elrab2 2871 . 2  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
) )
264, 22, 25sylanbrc 414 1  |-  ( (
ph  /\  s  <Q  r  /\  s  e.  ( 2nd `  L ) )  ->  r  e.  ( 2nd `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3966   -->wf 5167   ` cfv 5171  (class class class)co 5825   2ndc2nd 6088   Q.cnq 7201    +Q cplq 7203    <Q cltq 7206
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-eprel 4250  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-oadd 6368  df-omul 6369  df-er 6481  df-ec 6483  df-qs 6487  df-ni 7225  df-mi 7227  df-lti 7228  df-enq 7268  df-nqqs 7269  df-ltnqqs 7274
This theorem is referenced by:  cauappcvgprlemrnd  7571
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