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Theorem cauappcvgprlemopu 7424
Description: Lemma for cauappcvgpr 7438. 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 3903 . . . . . 6  |-  ( u  =  r  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  r
) )
21rexbidv 2415 . . . . 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 5392 . . . . . 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 7139 . . . . . . . 8  |-  Q.  e.  _V
65rabex 4042 . . . . . . 7  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
75rabex 4042 . . . . . . 7  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
86, 7op2nd 6013 . . . . . 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 2138 . . . . 5  |-  ( 2nd `  L )  =  {
u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }
102, 9elrab2 2816 . . . 4  |-  ( r  e.  ( 2nd `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
) )
1110simprbi 273 . . 3  |-  ( r  e.  ( 2nd `  L
)  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
)
1211adantl 275 . 2  |-  ( (
ph  /\  r  e.  ( 2nd `  L ) )  ->  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  r
)
13 simprr 506 . . . 4  |-  ( ( ( ph  /\  r  e.  ( 2nd `  L
) )  /\  (
q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  ->  ( ( F `
 q )  +Q  q )  <Q  r
)
14 ltbtwnnqq 7191 . . . 4  |-  ( ( ( F `  q
)  +Q  q ) 
<Q  r  <->  E. s  e.  Q.  ( ( ( F `
 q )  +Q  q )  <Q  s  /\  s  <Q  r ) )
1513, 14sylib 121 . . 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 506 . . . . . 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 504 . . . . . . 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 509 . . . . . . . . 9  |-  ( ( ( ( ph  /\  r  e.  ( 2nd `  L ) )  /\  ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  r ) )  /\  s  e.  Q. )  ->  q  e.  Q. )
1918adantr 274 . . . . . . . 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 505 . . . . . . . 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 2458 . . . . . . . 8  |-  ( ( q  e.  Q.  /\  ( ( F `  q )  +Q  q
)  <Q  s )  ->  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  s )
2219, 20, 21syl2anc 408 . . . . . . 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 3903 . . . . . . . . 9  |-  ( u  =  s  ->  (
( ( F `  q )  +Q  q
)  <Q  u  <->  ( ( F `  q )  +Q  q )  <Q  s
) )
2423rexbidv 2415 . . . . . . . 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 2816 . . . . . . 7  |-  ( s  e.  ( 2nd `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  s
) )
2617, 22, 25sylanbrc 413 . . . . . 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 304 . . . . 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 114 . . . 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 2511 . . 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 2531 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 103    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   -->wf 5089   ` cfv 5093  (class class class)co 5742   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058    <Q cltq 7061
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129
This theorem is referenced by:  cauappcvgprlemrnd  7426
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