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Theorem cauappcvgprlemcl 6941
Description: Lemma for cauappcvgpr 6950. 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 6933 . . 3  |-  ( ph  ->  ( E. s  e. 
Q.  s  e.  ( 1st `  L )  /\  E. r  e. 
Q.  r  e.  ( 2nd `  L ) ) )
6 ssrab2 3089 . . . . . 6  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  C_  Q.
7 nqex 6651 . . . . . . 7  |-  Q.  e.  _V
87elpw2 3953 . . . . . 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 144 . . . . 5  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  ~P Q.
10 ssrab2 3089 . . . . . 6  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  C_  Q.
117elpw2 3953 . . . . . 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 144 . . . . 5  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  ~P Q.
13 opelxpi 4423 . . . . 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 417 . . . 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 2155 . . 3  |-  L  e.  ( ~P Q.  X.  ~P Q. )
165, 15jctil 305 . 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 6938 . . 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 6939 . . 3  |-  ( ph  ->  A. s  e.  Q.  -.  ( s  e.  ( 1st `  L )  /\  s  e.  ( 2nd `  L ) ) )
191, 2, 3, 4cauappcvgprlemloc 6940 . . 3  |-  ( ph  ->  A. s  e.  Q.  A. r  e.  Q.  (
s  <Q  r  ->  (
s  e.  ( 1st `  L )  \/  r  e.  ( 2nd `  L
) ) ) )
2017, 18, 193jca 1119 . 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 6764 . 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 408 1  |-  ( ph  ->  L  e.  P. )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    /\ w3a 920    = wceq 1285    e. wcel 1434   A.wral 2353   E.wrex 2354   {crab 2357    C_ wss 2983   ~Pcpw 3401   <.cop 3420   class class class wbr 3806    X. cxp 4390   -->wf 4949   ` cfv 4953  (class class class)co 5564   1stc1st 5817   2ndc2nd 5818   Q.cnq 6568    +Q cplq 6570    <Q cltq 6573   P.cnp 6579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-irdg 6040  df-1o 6086  df-oadd 6090  df-omul 6091  df-er 6194  df-ec 6196  df-qs 6200  df-ni 6592  df-pli 6593  df-mi 6594  df-lti 6595  df-plpq 6632  df-mpq 6633  df-enq 6635  df-nqqs 6636  df-plqqs 6637  df-mqqs 6638  df-1nqqs 6639  df-rq 6640  df-ltnqqs 6641  df-inp 6754
This theorem is referenced by:  cauappcvgprlemladdfu  6942  cauappcvgprlemladdfl  6943  cauappcvgprlemladdru  6944  cauappcvgprlemladdrl  6945  cauappcvgprlemladd  6946  cauappcvgprlem1  6947  cauappcvgprlem2  6948  cauappcvgpr  6950
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