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Theorem icccmplem3 18345
Description: Lemma for icccmp 18346. (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 3271 . . . . . . . 8  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
42, 3eqsstri 3221 . . . . . . 7  |-  S  C_  ( A [,] B )
5 icccmp.5 . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
6 icccmp.6 . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
7 iccssre 10747 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
85, 6, 7syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
94, 8syl5ss 3203 . . . . . 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 18343 . . . . . . . 8  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
1615simpld 445 . . . . . . 7  |-  ( ph  ->  A  e.  S )
17 ne0i 3474 . . . . . . 7  |-  ( A  e.  S  ->  S  =/=  (/) )
1816, 17syl 15 . . . . . 6  |-  ( ph  ->  S  =/=  (/) )
1915simprd 449 . . . . . . 7  |-  ( ph  ->  A. y  e.  S  y  <_  B )
20 breq2 4043 . . . . . . . . 9  |-  ( v  =  B  ->  (
y  <_  v  <->  y  <_  B ) )
2120ralbidv 2576 . . . . . . . 8  |-  ( v  =  B  ->  ( A. y  e.  S  y  <_  v  <->  A. y  e.  S  y  <_  B ) )
2221rspcev 2897 . . . . . . 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 9730 . . . . . 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 9731 . . . . . 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 9734 . . . . . . 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 10731 . . . . . 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 3194 . . 3  |-  ( ph  ->  sup ( S ,  RR ,  <  )  e. 
U. U )
35 eluni2 3847 . . 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 3193 . . . . 5  |-  ( (
ph  /\  u  e.  U )  ->  u  e.  J )
3812rexmet 18313 . . . . . . 7  |-  D  e.  ( * Met `  RR )
39 eqid 2296 . . . . . . . . . 10  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
4012, 39tgioo 18318 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  D )
4110, 40eqtri 2316 . . . . . . . 8  |-  J  =  ( MetOpen `  D )
4241mopni2 18055 . . . . . . 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 2296 . . . . . . 7  |-  sup ( S ,  RR ,  <  )  =  sup ( S ,  RR ,  <  )
55 eqid 2296 . . . . . . 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 18344 . . . . . 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 2680 . . . 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 2680 . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   (/)c0 3468   ifcif 3578   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039    X. cxp 4703   ran crn 4706    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874   Fincfn 6879   supcsup 7209   RRcr 8752    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   2c2 9811   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   abscabs 11735   ↾t crest 13341   topGenctg 13358   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388
This theorem is referenced by:  icccmp  18346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-icc 10679  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655
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