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Theorem icccmplem3 21920
Description: Lemma for icccmp 21921. (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 3500 . . . . . . . 8  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
42, 3eqsstri 3448 . . . . . . 7  |-  S  C_  ( A [,] B )
5 icccmp.5 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
6 icccmp.6 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
7 iccssre 11741 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
85, 6, 7syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
94, 8syl5ss 3429 . . . . . 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 21918 . . . . . . . 8  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
1615simpld 466 . . . . . . 7  |-  ( ph  ->  A  e.  S )
17 ne0i 3728 . . . . . . 7  |-  ( A  e.  S  ->  S  =/=  (/) )
1816, 17syl 17 . . . . . 6  |-  ( ph  ->  S  =/=  (/) )
1915simprd 470 . . . . . . 7  |-  ( ph  ->  A. y  e.  S  y  <_  B )
20 breq2 4399 . . . . . . . . 9  |-  ( v  =  B  ->  (
y  <_  v  <->  y  <_  B ) )
2120ralbidv 2829 . . . . . . . 8  |-  ( v  =  B  ->  ( A. y  e.  S  y  <_  v  <->  A. y  e.  S  y  <_  B ) )
2221rspcev 3136 . . . . . . 7  |-  ( ( B  e.  RR  /\  A. y  e.  S  y  <_  B )  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
236, 19, 22syl2anc 673 . . . . . 6  |-  ( ph  ->  E. v  e.  RR  A. y  e.  S  y  <_  v )
24 suprcl 10591 . . . . . 6  |-  ( ( S  C_  RR  /\  S  =/=  (/)  /\  E. v  e.  RR  A. y  e.  S  y  <_  v
)  ->  sup ( S ,  RR ,  <  )  e.  RR )
259, 18, 23, 24syl3anc 1292 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  RR )
26 suprub 10592 . . . . . 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 1295 . . . . 5  |-  ( ph  ->  A  <_  sup ( S ,  RR ,  <  ) )
28 suprleub 10595 . . . . . . 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 1295 . . . . . 6  |-  ( ph  ->  ( sup ( S ,  RR ,  <  )  <_  B  <->  A. y  e.  S  y  <_  B ) )
3019, 29mpbird 240 . . . . 5  |-  ( ph  ->  sup ( S ,  RR ,  <  )  <_  B )
31 elicc2 11724 . . . . . 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 673 . . . . 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 1213 . . . 4  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e.  ( A [,] B
) )
341, 33sseldd 3419 . . 3  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e. 
U. U )
35 eluni2 4194 . . 3  |-  ( sup ( S ,  RR ,  <  )  e.  U. U 
<->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3634, 35sylib 201 . 2  |-  ( ph  ->  E. u  e.  U  sup ( S ,  RR ,  <  )  e.  u
)
3714sselda 3418 . . . . 5  |-  ( (
ph  /\  u  e.  U )  ->  u  e.  J )
3812rexmet 21887 . . . . . . 7  |-  D  e.  ( *Met `  RR )
39 eqid 2471 . . . . . . . . . 10  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4012, 39tgioo 21892 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  D )
4110, 40eqtri 2493 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
4241mopni2 21586 . . . . . . 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 1377 . . . . . 6  |-  ( ( u  e.  J  /\  sup ( S ,  RR ,  <  )  e.  u
)  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) ( ball `  D
) w )  C_  u )
4443ex 441 . . . . 5  |-  ( u  e.  J  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) (
ball `  D )
w )  C_  u
) )
4537, 44syl 17 . . . 4  |-  ( (
ph  /\  u  e.  U )  ->  ( sup ( S ,  RR ,  <  )  e.  u  ->  E. w  e.  RR+  ( sup ( S ,  RR ,  <  ) (
ball `  D )
w )  C_  u
) )
465ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  e.  RR )
476ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  RR )
4813ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  A  <_  B
)
4914ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  U  C_  J
)
501ad2antrr 740 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  ( A [,] B )  C_  U. U
)
51 simplr 770 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  u  e.  U
)
52 simprl 772 . . . . . 6  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  w  e.  RR+ )
53 simprr 774 . . . . . 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 2471 . . . . . 6  |-  sup ( S ,  RR ,  <  )  =  sup ( S ,  RR ,  <  )
55 eqid 2471 . . . . . 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 21919 . . . . 5  |-  ( ( ( ph  /\  u  e.  U )  /\  (
w  e.  RR+  /\  ( sup ( S ,  RR ,  <  ) ( ball `  D ) w ) 
C_  u ) )  ->  B  e.  S
)
5756rexlimdvaa 2872 . . . 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 2871 . 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   ~Pcpw 3942   U.cuni 4190   class class class wbr 4395    X. cxp 4837   ran crn 4840    |` cres 4841    o. ccom 4843   ` cfv 5589  (class class class)co 6308   Fincfn 7587   supcsup 7972   RRcr 9556    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880    / cdiv 10291   2c2 10681   RR+crp 11325   (,)cioo 11660   [,]cicc 11663   abscabs 13374   ↾t crest 15397   topGenctg 15414   *Metcxmt 19032   ballcbl 19034   MetOpencmopn 19037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-icc 11667  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000
This theorem is referenced by:  icccmp  21921
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