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Theorem cauappcvgprlemcl 7574
Description: Lemma for cauappcvgpr 7583. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-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
cauappcvgprlemcl  |-  ( ph  ->  L  e.  P. )
Distinct variable groups:    A, p    L, p, q    ph, p, q    F, l, u, p, q
Allowed substitution hints:    ph( u, l)    A( u, q, l)    L( u, l)

Proof of Theorem cauappcvgprlemcl
Dummy variables  r  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgpr.f . . . 4  |-  ( ph  ->  F : Q. --> Q. )
2 cauappcvgpr.app . . . 4  |-  ( 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 ) ) ) )
3 cauappcvgpr.bnd . . . 4  |-  ( ph  ->  A. p  e.  Q.  A  <Q  ( F `  p ) )
4 cauappcvgpr.lim . . . 4  |-  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 } >.
51, 2, 3, 4cauappcvgprlemm 7566 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
6 ssrab2 3213 . . . . . 6  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  C_  Q.
7 nqex 7284 . . . . . . 7  |-  Q.  e.  _V
87elpw2 4119 . . . . . 6  |-  ( { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }  e.  ~P Q.  <->  { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }  C_  Q. )
96, 8mpbir 145 . . . . 5  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  ~P Q.
10 ssrab2 3213 . . . . . 6  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  C_  Q.
117elpw2 4119 . . . . . 6  |-  ( { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }  e.  ~P Q.  <->  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `  q )  +Q  q
)  <Q  u }  C_  Q. )
1210, 11mpbir 145 . . . . 5  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  ~P Q.
13 opelxpi 4619 . . . . 5  |-  ( ( { l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }  e.  ~P Q.  /\  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  ~P Q. )  ->  <. { 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 } >.  e.  ( ~P Q.  X.  ~P Q. ) )
149, 12, 13mp2an 423 . . . 4  |-  <. { 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 } >.  e.  ( ~P Q.  X.  ~P Q. )
154, 14eqeltri 2230 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
165, 15jctil 310 . 2  |-  ( ph  ->  ( L  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. s  e.  Q.  s  e.  ( 1st `  L )  /\  E. r  e.  Q.  r  e.  ( 2nd `  L
) ) ) )
171, 2, 3, 4cauappcvgprlemrnd 7571 . . 3  |-  ( ph  ->  ( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) ) )
181, 2, 3, 4cauappcvgprlemdisj 7572 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
191, 2, 3, 4cauappcvgprlemloc 7573 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
2017, 18, 193jca 1162 . 2  |-  ( ph  ->  ( ( A. s  e.  Q.  ( s  e.  ( 1st `  L
)  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) )  /\  A. s  e.  Q.  -.  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  /\  A. s  e.  Q.  A. r  e.  Q.  ( s  <Q 
r  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) ) ) )
21 elnp1st2nd 7397 . 2  |-  ( L  e.  P.  <->  ( ( L  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )  /\  (
( A. s  e. 
Q.  ( s  e.  ( 1st `  L
)  <->  E. r  e.  Q.  ( s  <Q  r  /\  r  e.  ( 1st `  L ) ) )  /\  A. r  e.  Q.  ( r  e.  ( 2nd `  L
)  <->  E. s  e.  Q.  ( s  <Q  r  /\  s  e.  ( 2nd `  L ) ) ) )  /\  A. s  e.  Q.  -.  (
s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L
) )  /\  A. s  e.  Q.  A. r  e.  Q.  ( s  <Q 
r  ->  ( s  e.  ( 1st `  L
)  \/  r  e.  ( 2nd `  L
) ) ) ) ) )
2216, 20, 21sylanbrc 414 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1335    e. wcel 2128   A.wral 2435   E.wrex 2436   {crab 2439    C_ wss 3102   ~Pcpw 3543   <.cop 3563   class class class wbr 3966    X. cxp 4585   -->wf 5167   ` cfv 5171  (class class class)co 5825   1stc1st 6087   2ndc2nd 6088   Q.cnq 7201    +Q cplq 7203    <Q cltq 7206   P.cnp 7212
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-1o 6364  df-oadd 6368  df-omul 6369  df-er 6481  df-ec 6483  df-qs 6487  df-ni 7225  df-pli 7226  df-mi 7227  df-lti 7228  df-plpq 7265  df-mpq 7266  df-enq 7268  df-nqqs 7269  df-plqqs 7270  df-mqqs 7271  df-1nqqs 7272  df-rq 7273  df-ltnqqs 7274  df-inp 7387
This theorem is referenced by:  cauappcvgprlemladdfu  7575  cauappcvgprlemladdfl  7576  cauappcvgprlemladdru  7577  cauappcvgprlemladdrl  7578  cauappcvgprlemladd  7579  cauappcvgprlem1  7580  cauappcvgprlem2  7581  cauappcvgpr  7583
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