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Theorem icccmplem1 18327
Description: Lemma for icccmp 18330. (Contributed by Mario Carneiro, 18-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
icccmplem1  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
Distinct variable groups:    x, y,
z, B    ph, y    x, A, y, z    x, D   
x, T, z    z, J    y, S    x, U, y, z
Allowed substitution hints:    ph( x, z)    D( y, z)    S( x, z)    T( y)    J( x, y)

Proof of Theorem icccmplem1
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 icccmp.5 . . . . 5  |-  ( ph  ->  A  e.  RR )
21rexrd 8881 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 icccmp.6 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 8881 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 icccmp.7 . . . 4  |-  ( ph  ->  A  <_  B )
6 lbicc2 10752 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1182 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
8 icccmp.9 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
98, 7sseldd 3181 . . . . 5  |-  ( ph  ->  A  e.  U. U
)
10 eluni2 3831 . . . . 5  |-  ( A  e.  U. U  <->  E. u  e.  U  A  e.  u )
119, 10sylib 188 . . . 4  |-  ( ph  ->  E. u  e.  U  A  e.  u )
12 snssi 3759 . . . . . . . . . 10  |-  ( u  e.  U  ->  { u }  C_  U )
1312ad2antrl 708 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  C_  U )
14 snex 4216 . . . . . . . . . 10  |-  { u }  e.  _V
1514elpw 3631 . . . . . . . . 9  |-  ( { u }  e.  ~P U 
<->  { u }  C_  U )
1613, 15sylibr 203 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  ~P U )
17 snfi 6941 . . . . . . . . 9  |-  { u }  e.  Fin
1817a1i 10 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  Fin )
19 elin 3358 . . . . . . . 8  |-  ( { u }  e.  ( ~P U  i^i  Fin ) 
<->  ( { u }  e.  ~P U  /\  {
u }  e.  Fin ) )
2016, 18, 19sylanbrc 645 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  ( ~P U  i^i  Fin ) )
212adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  A  e.  RR* )
22 iccid 10701 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
2321, 22syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  =  { A } )
24 snssi 3759 . . . . . . . . 9  |-  ( A  e.  u  ->  { A }  C_  u )
2524ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { A }  C_  u
)
2623, 25eqsstrd 3212 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  C_  u )
27 unieq 3836 . . . . . . . . . 10  |-  ( z  =  { u }  ->  U. z  =  U. { u } )
28 vex 2791 . . . . . . . . . . 11  |-  u  e. 
_V
2928unisn 3843 . . . . . . . . . 10  |-  U. {
u }  =  u
3027, 29syl6eq 2331 . . . . . . . . 9  |-  ( z  =  { u }  ->  U. z  =  u )
3130sseq2d 3206 . . . . . . . 8  |-  ( z  =  { u }  ->  ( ( A [,] A )  C_  U. z  <->  ( A [,] A ) 
C_  u ) )
3231rspcev 2884 . . . . . . 7  |-  ( ( { u }  e.  ( ~P U  i^i  Fin )  /\  ( A [,] A )  C_  u
)  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
)
3320, 26, 32syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  E. z  e.  ( ~P U  i^i  Fin )
( A [,] A
)  C_  U. z
)
3433expr 598 . . . . 5  |-  ( (
ph  /\  u  e.  U )  ->  ( A  e.  u  ->  E. z  e.  ( ~P U  i^i  Fin )
( A [,] A
)  C_  U. z
) )
3534rexlimdva 2667 . . . 4  |-  ( ph  ->  ( E. u  e.  U  A  e.  u  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
) )
3611, 35mpd 14 . . 3  |-  ( ph  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
)
37 oveq2 5866 . . . . . 6  |-  ( x  =  A  ->  ( A [,] x )  =  ( A [,] A
) )
3837sseq1d 3205 . . . . 5  |-  ( x  =  A  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] A ) 
C_  U. z ) )
3938rexbidv 2564 . . . 4  |-  ( x  =  A  ->  ( E. z  e.  ( ~P U  i^i  Fin )
( A [,] x
)  C_  U. z  <->  E. z  e.  ( ~P U  i^i  Fin )
( A [,] A
)  C_  U. z
) )
40 icccmp.4 . . . 4  |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }
4139, 40elrab2 2925 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
) )
427, 36, 41sylanbrc 645 . 2  |-  ( ph  ->  A  e.  S )
43 ssrab2 3258 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
4440, 43eqsstri 3208 . . . . 5  |-  S  C_  ( A [,] B )
4544sseli 3176 . . . 4  |-  ( y  e.  S  ->  y  e.  ( A [,] B
) )
46 elicc2 10715 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
471, 3, 46syl2anc 642 . . . . . 6  |-  ( ph  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
4847biimpa 470 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) )
4948simp3d 969 . . . 4  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  <_  B )
5045, 49sylan2 460 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  y  <_  B )
5150ralrimiva 2626 . 2  |-  ( ph  ->  A. y  e.  S  y  <_  B )
5242, 51jca 518 1  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   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   RRcr 8736   RR*cxr 8866    <_ cle 8868    - cmin 9037   (,)cioo 10656   [,]cicc 10659   abscabs 11719   ↾t crest 13325   topGenctg 13342
This theorem is referenced by:  icccmplem2  18328  icccmplem3  18329
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-pre-lttri 8811  ax-pre-lttrn 8812
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-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-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-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663
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