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Theorem icccmplem1 18343
Description: Lemma for icccmp 18346. (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 8897 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 icccmp.6 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 8897 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 icccmp.7 . . . 4  |-  ( ph  ->  A  <_  B )
6 lbicc2 10768 . . . 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 3194 . . . . 5  |-  ( ph  ->  A  e.  U. U
)
10 eluni2 3847 . . . . 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 3775 . . . . . . . . . 10  |-  ( u  e.  U  ->  { u }  C_  U )
1312ad2antrl 708 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  C_  U )
14 snex 4232 . . . . . . . . . 10  |-  { u }  e.  _V
1514elpw 3644 . . . . . . . . 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 6957 . . . . . . . . 9  |-  { u }  e.  Fin
1817a1i 10 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  Fin )
19 elin 3371 . . . . . . . 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 10717 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
2321, 22syl 15 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  =  { A } )
24 snssi 3775 . . . . . . . . 9  |-  ( A  e.  u  ->  { A }  C_  u )
2524ad2antll 709 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { A }  C_  u
)
2623, 25eqsstrd 3225 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  C_  u )
27 unieq 3852 . . . . . . . . . 10  |-  ( z  =  { u }  ->  U. z  =  U. { u } )
28 vex 2804 . . . . . . . . . . 11  |-  u  e. 
_V
2928unisn 3859 . . . . . . . . . 10  |-  U. {
u }  =  u
3027, 29syl6eq 2344 . . . . . . . . 9  |-  ( z  =  { u }  ->  U. z  =  u )
3130sseq2d 3219 . . . . . . . 8  |-  ( z  =  { u }  ->  ( ( A [,] A )  C_  U. z  <->  ( A [,] A ) 
C_  u ) )
3231rspcev 2897 . . . . . . 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 2680 . . . 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 5882 . . . . . 6  |-  ( x  =  A  ->  ( A [,] x )  =  ( A [,] A
) )
3837sseq1d 3218 . . . . 5  |-  ( x  =  A  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] A ) 
C_  U. z ) )
3938rexbidv 2577 . . . 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 2938 . . 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 3271 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
4440, 43eqsstri 3221 . . . . 5  |-  S  C_  ( A [,] B )
4544sseli 3189 . . . 4  |-  ( y  e.  S  ->  y  e.  ( A [,] B
) )
46 elicc2 10731 . . . . . . 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 2639 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   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   RRcr 8752   RR*cxr 8882    <_ cle 8884    - cmin 9053   (,)cioo 10672   [,]cicc 10675   abscabs 11735   ↾t crest 13341   topGenctg 13358
This theorem is referenced by:  icccmplem2  18344  icccmplem3  18345
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-pre-lttri 8827  ax-pre-lttrn 8828
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-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-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-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-icc 10679
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