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Theorem cauappcvgprlemopu 7622
Description: Lemma for cauappcvgpr 7636. The upper 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
cauappcvgprlemopu  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
r  /\  s  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 cauappcvgprlemopu
StepHypRef Expression
1 breq2 4002 . . . . . 6  |-  ( u  =  r  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  r
) )
21rexbidv 2476 . . . . 5  |-  ( u  =  r  ->  ( E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
) )
3 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 } >.
43fveq2i 5510 . . . . . 6  |-  ( 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 } >. )
5 nqex 7337 . . . . . . . 8  |-  Q.  e.  _V
65rabex 4142 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
75rabex 4142 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
86, 7op2nd 6138 . . . . . 6  |-  ( 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 }
94, 8eqtri 2196 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
102, 9elrab2 2894 . . . 4  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
) )
1110simprbi 275 . . 3  |-  ( r  e.  ( 2nd `  L
)  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
)
1211adantl 277 . 2  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
)
13 simprr 531 . . . 4  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  ->  ( ( F `
 q )  +Q  q )  <Q  r
)
14 ltbtwnnqq 7389 . . . 4  |-  ( ( ( F `  q
)  +Q  q ) 
<Q  r  <->  E. s  e.  Q.  ( ( ( F `
 q )  +Q  q )  <Q  s  /\  s  <Q  r ) )
1513, 14sylib 122 . . 3  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  ->  E. s  e.  Q.  ( ( ( F `
 q )  +Q  q )  <Q  s  /\  s  <Q  r ) )
16 simprr 531 . . . . . 6  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  s  <Q  r )
17 simplr 528 . . . . . . 7  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  s  e.  Q. )
18 simplrl 535 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  ->  q  e.  Q. )
1918adantr 276 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  q  e.  Q. )
20 simprl 529 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  (
( F `  q
)  +Q  q ) 
<Q  s )
21 rspe 2524 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  s )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  s )
2219, 20, 21syl2anc 411 . . . . . . 7  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
)
23 breq2 4002 . . . . . . . . 9  |-  ( u  =  s  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  s
) )
2423rexbidv 2476 . . . . . . . 8  |-  ( u  =  s  ->  ( E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u  <->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
2524, 9elrab2 2894 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
2617, 22, 25sylanbrc 417 . . . . . 6  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  s  e.  ( 2nd `  L
) )
2716, 26jca 306 . . . . 5  |-  ( ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  /\  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r ) )  ->  (
s  <Q  r  /\  s  e.  ( 2nd `  L
) ) )
2827ex 115 . . . 4  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  ->  ( ( ( ( F `  q
)  +Q  q ) 
<Q  s  /\  s  <Q  r )  ->  (
s  <Q  r  /\  s  e.  ( 2nd `  L
) ) ) )
2928reximdva 2577 . . 3  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  ->  ( E. s  e.  Q.  ( ( ( F `  q )  +Q  q )  <Q 
s  /\  s  <Q  r )  ->  E. s  e.  Q.  ( s  <Q 
r  /\  s  e.  ( 2nd `  L ) ) ) )
3015, 29mpd 13 . 2  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  ->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) )
3112, 30rexlimddv 2597 1  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. s  e.  Q.  ( s  <Q 
r  /\  s  e.  ( 2nd `  L ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2146   A.wral 2453   E.wrex 2454   {crab 2457   <.cop 3592   class class class wbr 3998   -->wf 5204   ` cfv 5208  (class class class)co 5865   2ndc2nd 6130   Q.cnq 7254    +Q cplq 7256    <Q cltq 7259
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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327
This theorem is referenced by:  cauappcvgprlemrnd  7624
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