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Theorem icccmplem3 18329
Description: Lemma for icccmp 18330. (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 3258 . . . . . . . 8  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
42, 3eqsstri 3208 . . . . . . 7  |-  S  C_  ( A [,] B )
5 icccmp.5 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
6 icccmp.6 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
7 iccssre 10731 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
85, 6, 7syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
94, 8syl5ss 3190 . . . . . 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 18327 . . . . . . . 8  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
1615simpld 445 . . . . . . 7  |-  ( ph  ->  A  e.  S )
17 ne0i 3461 . . . . . . 7  |-  ( A  e.  S  ->  S  =/=  (/) )
1816, 17syl 15 . . . . . 6  |-  ( ph  ->  S  =/=  (/) )
1915simprd 449 . . . . . . 7  |-  ( ph  ->  A. y  e.  S  y  <_  B )
20 breq2 4027 . . . . . . . . 9  |-  ( v  =  B  ->  (
y  <_  v  <->  y  <_  B ) )
2120ralbidv 2563 . . . . . . . 8  |-  ( v  =  B  ->  ( A. y  e.  S  y  <_  v  <->  A. y  e.  S  y  <_  B ) )
2221rspcev 2884 . . . . . . 7  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
236, 19, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
24 suprcl 9714 . . . . . 6  |-  ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. v  e.  RR  A. y  e.  S  y  <_  v
)  ->  sup ( S ,  RR ,  <  )  e.  RR )
259, 18, 23, 24syl3anc 1182 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  RR )
26 suprub 9715 . . . . . 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 1185 . . . . 5  |-  ( ph  ->  A  <_  sup ( S ,  RR ,  <  ) )
28 suprleub 9718 . . . . . . 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 1185 . . . . . 6  |-  ( ph  ->  ( sup ( S ,  RR ,  <  )  <_  B  <->  A. y  e.  S  y  <_  B ) )
3019, 29mpbird 223 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  <_  B )
31 elicc2 10715 . . . . . 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 642 . . . . 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 1135 . . . 4  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  ( A [,] B
) )
341, 33sseldd 3181 . . 3  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e. 
U. U )
35 eluni2 3831 . . 3  |-  ( sup ( S ,  RR ,  <  )  e.  U. U 
<->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3634, 35sylib 188 . 2  |-  ( ph  ->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3714sselda 3180 . . . . 5  |-  ( (
ph  /\  u  e.  U )  ->  u  e.  J )
3812rexmet 18297 . . . . . . 7  |-  D  e.  ( * Met `  RR )
39 eqid 2283 . . . . . . . . . 10  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4012, 39tgioo 18302 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  D )
4110, 40eqtri 2303 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
4241mopni2 18039 . . . . . . 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 1264 . . . . . 6  |-  ( ( u  e.  J  /\  sup ( S ,  RR ,  <  )  e.  u
)  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u )
4443ex 423 . . . . 5  |-  ( u  e.  J  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) (
ball `  D )
w )  C_  u
) )
4537, 44syl 15 . . . 4  |-  ( (
ph  /\  u  e.  U )  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) (
ball `  D )
w )  C_  u
) )
465ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  e.  RR )
476ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  RR )
4813ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  <_  B
)
4914ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  U  C_  J
)
501ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  ( A [,] B )  C_  U. U
)
51 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  u  e.  U
)
52 simprl 732 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  w  e.  RR+ )
53 simprr 733 . . . . . . 7  |-  ( ( ( 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 2283 . . . . . . 7  |-  sup ( S ,  RR ,  <  )  =  sup ( S ,  RR ,  <  )
55 eqid 2283 . . . . . . 7  |-  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 18328 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  S
)
5756expr 598 . . . . 5  |-  ( ( ( ph  /\  u  e.  U )  /\  w  e.  RR+ )  ->  (
( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u  ->  B  e.  S
) )
5857rexlimdva 2667 . . . 4  |-  ( (
ph  /\  u  e.  U )  ->  ( E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u  ->  B  e.  S ) )
5945, 58syld 40 . . 3  |-  ( (
ph  /\  u  e.  U )  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  B  e.  S ) )
6059rexlimdva 2667 . 2  |-  ( ph  ->  ( E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u  ->  B  e.  S ) )
6136, 60mpd 14 1  |-  ( ph  ->  B  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023    X. cxp 4687   ran crn 4690    |` cres 4691    o. ccom 4693   ` cfv 5255  (class class class)co 5858   Fincfn 6863   supcsup 7193   RRcr 8736    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   abscabs 11719   ↾t crest 13325   topGenctg 13342   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372
This theorem is referenced by:  icccmp  18330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639
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