ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cauappcvgprlemlol Unicode version

Theorem cauappcvgprlemlol 7206
Description: Lemma for cauappcvgpr 7221. The lower cut of the putative limit is lower. (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
cauappcvgprlemlol  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  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 cauappcvgprlemlol
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelnq 6924 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4490 . . . 4  |-  ( s 
<Q  r  ->  ( s  e.  Q.  /\  r  e.  Q. ) )
32simpld 110 . . 3  |-  ( s 
<Q  r  ->  s  e. 
Q. )
433ad2ant2 965 . 2  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  Q. )
5 oveq1 5659 . . . . . . . 8  |-  ( l  =  r  ->  (
l  +Q  q )  =  ( r  +Q  q ) )
65breq1d 3855 . . . . . . 7  |-  ( l  =  r  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( r  +Q  q )  <Q  ( F `  q )
) )
76rexbidv 2381 . . . . . 6  |-  ( l  =  r  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
8 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 } >.
98fveq2i 5308 . . . . . . 7  |-  ( 1st `  L )  =  ( 1st `  <. { 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 } >. )
10 nqex 6922 . . . . . . . . 9  |-  Q.  e.  _V
1110rabex 3983 . . . . . . . 8  |-  { l  e.  Q.  |  E. q  e.  Q.  (
l  +Q  q ) 
<Q  ( F `  q
) }  e.  _V
1210rabex 3983 . . . . . . . 8  |-  { u  e.  Q.  |  E. q  e.  Q.  ( ( F `
 q )  +Q  q )  <Q  u }  e.  _V
1311, 12op1st 5917 . . . . . . 7  |-  ( 1st `  <. { 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 } >. )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
149, 13eqtri 2108 . . . . . 6  |-  ( 1st `  L )  =  {
l  e.  Q.  |  E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q ) }
157, 14elrab2 2774 . . . . 5  |-  ( r  e.  ( 1st `  L
)  <->  ( r  e. 
Q.  /\  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
) )
1615simprbi 269 . . . 4  |-  ( r  e.  ( 1st `  L
)  ->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
)
17163ad2ant3 966 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( r  +Q  q )  <Q  ( F `  q )
)
18 simpll2 983 . . . . . . 7  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  s  <Q  r )
19 ltanqg 6959 . . . . . . . . 9  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2019adantl 271 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  /\  q  e.  Q. )  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
214ad2antrr 472 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  s  e.  Q. )
222simprd 112 . . . . . . . . . 10  |-  ( s 
<Q  r  ->  r  e. 
Q. )
23223ad2ant2 965 . . . . . . . . 9  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  r  e.  Q. )
2423ad2antrr 472 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  r  e.  Q. )
25 simplr 497 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  q  e.  Q. )
26 addcomnqg 6940 . . . . . . . . 9  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
2726adantl 271 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  /\  q  e.  Q. )  /\  ( r  +Q  q
)  <Q  ( F `  q ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
2820, 21, 24, 25, 27caovord2d 5814 . . . . . . 7  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  <Q  r  <->  ( s  +Q  q )  <Q  (
r  +Q  q ) ) )
2918, 28mpbid 145 . . . . . 6  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( r  +Q  q
) )
30 ltsonq 6957 . . . . . . 7  |-  <Q  Or  Q.
3130, 1sotri 4827 . . . . . 6  |-  ( ( ( s  +Q  q
)  <Q  ( r  +Q  q )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) )
3229, 31sylancom 411 . . . . 5  |-  ( ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  /\  (
r  +Q  q ) 
<Q  ( F `  q
) )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) )
3332ex 113 . . . 4  |-  ( ( ( ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L
) )  /\  q  e.  Q. )  ->  (
( r  +Q  q
)  <Q  ( F `  q )  ->  (
s  +Q  q ) 
<Q  ( F `  q
) ) )
3433reximdva 2475 . . 3  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  ( E. q  e.  Q.  (
r  +Q  q ) 
<Q  ( F `  q
)  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
3517, 34mpd 13 . 2  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
)
36 oveq1 5659 . . . . 5  |-  ( l  =  s  ->  (
l  +Q  q )  =  ( s  +Q  q ) )
3736breq1d 3855 . . . 4  |-  ( l  =  s  ->  (
( l  +Q  q
)  <Q  ( F `  q )  <->  ( s  +Q  q )  <Q  ( F `  q )
) )
3837rexbidv 2381 . . 3  |-  ( l  =  s  ->  ( E. q  e.  Q.  ( l  +Q  q
)  <Q  ( F `  q )  <->  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
3938, 14elrab2 2774 . 2  |-  ( s  e.  ( 1st `  L
)  <->  ( s  e. 
Q.  /\  E. q  e.  Q.  ( s  +Q  q )  <Q  ( F `  q )
) )
404, 35, 39sylanbrc 408 1  |-  ( (
ph  /\  s  <Q  r  /\  r  e.  ( 1st `  L ) )  ->  s  e.  ( 1st `  L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360   {crab 2363   <.cop 3449   class class class wbr 3845   -->wf 5011   ` cfv 5015  (class class class)co 5652   1stc1st 5909   Q.cnq 6839    +Q cplq 6841    <Q cltq 6844
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-eprel 4116  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-irdg 6135  df-oadd 6185  df-omul 6186  df-er 6292  df-ec 6294  df-qs 6298  df-ni 6863  df-pli 6864  df-mi 6865  df-lti 6866  df-plpq 6903  df-enq 6906  df-nqqs 6907  df-plqqs 6908  df-ltnqqs 6912
This theorem is referenced by:  cauappcvgprlemrnd  7209
  Copyright terms: Public domain W3C validator