MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  icccmplem3 Structured version   Unicode version

Theorem icccmplem3 18845
Description: Lemma for icccmp 18846. (Contributed by Mario Carneiro, 13-Jun-2014.)
Hypotheses
Ref Expression
icccmp.1  |-  J  =  ( topGen `  ran  (,) )
icccmp.2  |-  T  =  ( Jt  ( A [,] B ) )
icccmp.3  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
icccmp.4  |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }
icccmp.5  |-  ( ph  ->  A  e.  RR )
icccmp.6  |-  ( ph  ->  B  e.  RR )
icccmp.7  |-  ( ph  ->  A  <_  B )
icccmp.8  |-  ( ph  ->  U  C_  J )
icccmp.9  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
Assertion
Ref Expression
icccmplem3  |-  ( ph  ->  B  e.  S )
Distinct variable groups:    x, z, B    x, A, z    x, D    x, T, z    z, J    x, U, z
Allowed substitution hints:    ph( x, z)    D( z)    S( x, z)    J( x)

Proof of Theorem icccmplem3
Dummy variables  u  v  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 icccmp.9 . . . 4  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
2 icccmp.4 . . . . . . . 8  |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }
3 ssrab2 3420 . . . . . . . 8  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
42, 3eqsstri 3370 . . . . . . 7  |-  S  C_  ( A [,] B )
5 icccmp.5 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
6 icccmp.6 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
7 iccssre 10982 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
85, 6, 7syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
94, 8syl5ss 3351 . . . . . 6  |-  ( ph  ->  S  C_  RR )
10 icccmp.1 . . . . . . . . 9  |-  J  =  ( topGen `  ran  (,) )
11 icccmp.2 . . . . . . . . 9  |-  T  =  ( Jt  ( A [,] B ) )
12 icccmp.3 . . . . . . . . 9  |-  D  =  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )
13 icccmp.7 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
14 icccmp.8 . . . . . . . . 9  |-  ( ph  ->  U  C_  J )
1510, 11, 12, 2, 5, 6, 13, 14, 1icccmplem1 18843 . . . . . . . 8  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
1615simpld 446 . . . . . . 7  |-  ( ph  ->  A  e.  S )
17 ne0i 3626 . . . . . . 7  |-  ( A  e.  S  ->  S  =/=  (/) )
1816, 17syl 16 . . . . . 6  |-  ( ph  ->  S  =/=  (/) )
1915simprd 450 . . . . . . 7  |-  ( ph  ->  A. y  e.  S  y  <_  B )
20 breq2 4208 . . . . . . . . 9  |-  ( v  =  B  ->  (
y  <_  v  <->  y  <_  B ) )
2120ralbidv 2717 . . . . . . . 8  |-  ( v  =  B  ->  ( A. y  e.  S  y  <_  v  <->  A. y  e.  S  y  <_  B ) )
2221rspcev 3044 . . . . . . 7  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
236, 19, 22syl2anc 643 . . . . . 6  |-  ( ph  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
24 suprcl 9958 . . . . . 6  |-  ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. v  e.  RR  A. y  e.  S  y  <_  v
)  ->  sup ( S ,  RR ,  <  )  e.  RR )
259, 18, 23, 24syl3anc 1184 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  RR )
26 suprub 9959 . . . . . 6  |-  ( ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. v  e.  RR  A. y  e.  S  y  <_  v )  /\  A  e.  S )  ->  A  <_  sup ( S ,  RR ,  <  ) )
279, 18, 23, 16, 26syl31anc 1187 . . . . 5  |-  ( ph  ->  A  <_  sup ( S ,  RR ,  <  ) )
28 suprleub 9962 . . . . . . 7  |-  ( ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. v  e.  RR  A. y  e.  S  y  <_  v )  /\  B  e.  RR )  ->  ( sup ( S ,  RR ,  <  )  <_  B  <->  A. y  e.  S  y  <_  B ) )
299, 18, 23, 6, 28syl31anc 1187 . . . . . 6  |-  ( ph  ->  ( sup ( S ,  RR ,  <  )  <_  B  <->  A. y  e.  S  y  <_  B ) )
3019, 29mpbird 224 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  <_  B )
31 elicc2 10965 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( sup ( S ,  RR ,  <  )  e.  ( A [,] B )  <->  ( sup ( S ,  RR ,  <  )  e.  RR  /\  A  <_  sup ( S ,  RR ,  <  )  /\  sup ( S ,  RR ,  <  )  <_  B
) ) )
325, 6, 31syl2anc 643 . . . . 5  |-  ( ph  ->  ( sup ( S ,  RR ,  <  )  e.  ( A [,] B )  <->  ( sup ( S ,  RR ,  <  )  e.  RR  /\  A  <_  sup ( S ,  RR ,  <  )  /\  sup ( S ,  RR ,  <  )  <_  B
) ) )
3325, 27, 30, 32mpbir3and 1137 . . . 4  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  ( A [,] B
) )
341, 33sseldd 3341 . . 3  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e. 
U. U )
35 eluni2 4011 . . 3  |-  ( sup ( S ,  RR ,  <  )  e.  U. U 
<->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3634, 35sylib 189 . 2  |-  ( ph  ->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3714sselda 3340 . . . . 5  |-  ( (
ph  /\  u  e.  U )  ->  u  e.  J )
3812rexmet 18812 . . . . . . 7  |-  D  e.  ( * Met `  RR )
39 eqid 2435 . . . . . . . . . 10  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4012, 39tgioo 18817 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  D )
4110, 40eqtri 2455 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
4241mopni2 18513 . . . . . . 7  |-  ( ( D  e.  ( * Met `  RR )  /\  u  e.  J  /\  sup ( S ,  RR ,  <  )  e.  u )  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u )
4338, 42mp3an1 1266 . . . . . 6  |-  ( ( u  e.  J  /\  sup ( S ,  RR ,  <  )  e.  u
)  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u )
4443ex 424 . . . . 5  |-  ( u  e.  J  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) (
ball `  D )
w )  C_  u
) )
4537, 44syl 16 . . . 4  |-  ( (
ph  /\  u  e.  U )  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) (
ball `  D )
w )  C_  u
) )
465ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  e.  RR )
476ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  RR )
4813ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  <_  B
)
4914ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  U  C_  J
)
501ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  ( A [,] B )  C_  U. U
)
51 simplr 732 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  u  e.  U
)
52 simprl 733 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  w  e.  RR+ )
53 simprr 734 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  ( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u )
54 eqid 2435 . . . . . 6  |-  sup ( S ,  RR ,  <  )  =  sup ( S ,  RR ,  <  )
55 eqid 2435 . . . . . 6  |-  if ( ( sup ( S ,  RR ,  <  )  +  ( w  / 
2 ) )  <_  B ,  ( sup ( S ,  RR ,  <  )  +  ( w  /  2 ) ) ,  B )  =  if ( ( sup ( S ,  RR ,  <  )  +  ( w  /  2 ) )  <_  B , 
( sup ( S ,  RR ,  <  )  +  ( w  / 
2 ) ) ,  B )
5610, 11, 12, 2, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55icccmplem2 18844 . . . . 5  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  S
)
5756rexlimdvaa 2823 . . . 4  |-  ( (
ph  /\  u  e.  U )  ->  ( E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u  ->  B  e.  S ) )
5845, 57syld 42 . . 3  |-  ( (
ph  /\  u  e.  U )  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  B  e.  S ) )
5958rexlimdva 2822 . 2  |-  ( ph  ->  ( E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u  ->  B  e.  S ) )
6036, 59mpd 15 1  |-  ( ph  ->  B  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791   U.cuni 4007   class class class wbr 4204    X. cxp 4868   ran crn 4871    |` cres 4872    o. ccom 4874   ` cfv 5446  (class class class)co 6073   Fincfn 7101   supcsup 7437   RRcr 8979    + caddc 8983    < clt 9110    <_ cle 9111    - cmin 9281    / cdiv 9667   2c2 10039   RR+crp 10602   (,)cioo 10906   [,]cicc 10909   abscabs 12029   ↾t crest 13638   topGenctg 13655   * Metcxmt 16676   ballcbl 16678   MetOpencmopn 16681
This theorem is referenced by:  icccmp  18846
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-q 10565  df-rp 10603  df-xneg 10700  df-xadd 10701  df-xmul 10702  df-ioo 10910  df-icc 10913  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-topgen 13657  df-psmet 16684  df-xmet 16685  df-met 16686  df-bl 16687  df-mopn 16688  df-top 16953  df-bases 16955  df-topon 16956
  Copyright terms: Public domain W3C validator