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Theorem icccmplem3 20381
Description: Lemma for icccmp 20382. (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 3432 . . . . . . . 8  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
42, 3eqsstri 3381 . . . . . . 7  |-  S  C_  ( A [,] B )
5 icccmp.5 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
6 icccmp.6 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
7 iccssre 11369 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
85, 6, 7syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
94, 8syl5ss 3362 . . . . . 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 20379 . . . . . . . 8  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
1615simpld 459 . . . . . . 7  |-  ( ph  ->  A  e.  S )
17 ne0i 3638 . . . . . . 7  |-  ( A  e.  S  ->  S  =/=  (/) )
1816, 17syl 16 . . . . . 6  |-  ( ph  ->  S  =/=  (/) )
1915simprd 463 . . . . . . 7  |-  ( ph  ->  A. y  e.  S  y  <_  B )
20 breq2 4291 . . . . . . . . 9  |-  ( v  =  B  ->  (
y  <_  v  <->  y  <_  B ) )
2120ralbidv 2730 . . . . . . . 8  |-  ( v  =  B  ->  ( A. y  e.  S  y  <_  v  <->  A. y  e.  S  y  <_  B ) )
2221rspcev 3068 . . . . . . 7  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
236, 19, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
24 suprcl 10282 . . . . . 6  |-  ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. v  e.  RR  A. y  e.  S  y  <_  v
)  ->  sup ( S ,  RR ,  <  )  e.  RR )
259, 18, 23, 24syl3anc 1218 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  RR )
26 suprub 10283 . . . . . 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 1221 . . . . 5  |-  ( ph  ->  A  <_  sup ( S ,  RR ,  <  ) )
28 suprleub 10286 . . . . . . 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 1221 . . . . . 6  |-  ( ph  ->  ( sup ( S ,  RR ,  <  )  <_  B  <->  A. y  e.  S  y  <_  B ) )
3019, 29mpbird 232 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  <_  B )
31 elicc2 11352 . . . . . 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 661 . . . . 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 1171 . . . 4  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  ( A [,] B
) )
341, 33sseldd 3352 . . 3  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e. 
U. U )
35 eluni2 4090 . . 3  |-  ( sup ( S ,  RR ,  <  )  e.  U. U 
<->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3634, 35sylib 196 . 2  |-  ( ph  ->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3714sselda 3351 . . . . 5  |-  ( (
ph  /\  u  e.  U )  ->  u  e.  J )
3812rexmet 20348 . . . . . . 7  |-  D  e.  ( *Met `  RR )
39 eqid 2438 . . . . . . . . . 10  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4012, 39tgioo 20353 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  D )
4110, 40eqtri 2458 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
4241mopni2 20048 . . . . . . 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 1301 . . . . . 6  |-  ( ( u  e.  J  /\  sup ( S ,  RR ,  <  )  e.  u
)  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u )
4443ex 434 . . . . 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 725 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  e.  RR )
476ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  RR )
4813ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  <_  B
)
4914ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  U  C_  J
)
501ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  ( A [,] B )  C_  U. U
)
51 simplr 754 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  u  e.  U
)
52 simprl 755 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  w  e.  RR+ )
53 simprr 756 . . . . . 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 2438 . . . . . 6  |-  sup ( S ,  RR ,  <  )  =  sup ( S ,  RR ,  <  )
55 eqid 2438 . . . . . 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 20380 . . . . 5  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  S
)
5756rexlimdvaa 2837 . . . 4  |-  ( (
ph  /\  u  e.  U )  ->  ( E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u  ->  B  e.  S ) )
5845, 57syld 44 . . 3  |-  ( (
ph  /\  u  e.  U )  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  B  e.  S ) )
5958rexlimdva 2836 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711   {crab 2714    i^i cin 3322    C_ wss 3323   (/)c0 3632   ifcif 3786   ~Pcpw 3855   U.cuni 4086   class class class wbr 4287    X. cxp 4833   ran crn 4836    |` cres 4837    o. ccom 4839   ` cfv 5413  (class class class)co 6086   Fincfn 7302   supcsup 7682   RRcr 9273    + caddc 9277    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   2c2 10363   RR+crp 10983   (,)cioo 11292   [,]cicc 11295   abscabs 12715   ↾t crest 14351   topGenctg 14368   *Metcxmt 17781   ballcbl 17783   MetOpencmopn 17786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-icc 11299  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-topgen 14374  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-top 18483  df-bases 18485  df-topon 18486
This theorem is referenced by:  icccmp  20382
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