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Theorem icccmplem1 21918
Description: Lemma for icccmp 21921. (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 9708 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 icccmp.6 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 9708 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 icccmp.7 . . . 4  |-  ( ph  ->  A  <_  B )
6 lbicc2 11774 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1292 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
8 icccmp.9 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
98, 7sseldd 3419 . . . . 5  |-  ( ph  ->  A  e.  U. U
)
10 eluni2 4194 . . . . 5  |-  ( A  e.  U. U  <->  E. u  e.  U  A  e.  u )
119, 10sylib 201 . . . 4  |-  ( ph  ->  E. u  e.  U  A  e.  u )
12 snssi 4107 . . . . . . . 8  |-  ( u  e.  U  ->  { u }  C_  U )
1312ad2antrl 742 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  C_  U )
14 snex 4641 . . . . . . . 8  |-  { u }  e.  _V
1514elpw 3948 . . . . . . 7  |-  ( { u }  e.  ~P U 
<->  { u }  C_  U )
1613, 15sylibr 217 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  ~P U )
17 snfi 7668 . . . . . . 7  |-  { u }  e.  Fin
1817a1i 11 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  Fin )
1916, 18elind 3609 . . . . 5  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  ( ~P U  i^i  Fin ) )
202adantr 472 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  A  e.  RR* )
21 iccid 11706 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
2220, 21syl 17 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  =  { A } )
23 snssi 4107 . . . . . . 7  |-  ( A  e.  u  ->  { A }  C_  u )
2423ad2antll 743 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { A }  C_  u
)
2522, 24eqsstrd 3452 . . . . 5  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  C_  u )
26 unieq 4198 . . . . . . . 8  |-  ( z  =  { u }  ->  U. z  =  U. { u } )
27 vex 3034 . . . . . . . . 9  |-  u  e. 
_V
2827unisn 4205 . . . . . . . 8  |-  U. {
u }  =  u
2926, 28syl6eq 2521 . . . . . . 7  |-  ( z  =  { u }  ->  U. z  =  u )
3029sseq2d 3446 . . . . . 6  |-  ( z  =  { u }  ->  ( ( A [,] A )  C_  U. z  <->  ( A [,] A ) 
C_  u ) )
3130rspcev 3136 . . . . 5  |-  ( ( { u }  e.  ( ~P U  i^i  Fin )  /\  ( A [,] A )  C_  u
)  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
)
3219, 25, 31syl2anc 673 . . . 4  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  E. z  e.  ( ~P U  i^i  Fin )
( A [,] A
)  C_  U. z
)
3311, 32rexlimddv 2875 . . 3  |-  ( ph  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
)
34 oveq2 6316 . . . . . 6  |-  ( x  =  A  ->  ( A [,] x )  =  ( A [,] A
) )
3534sseq1d 3445 . . . . 5  |-  ( x  =  A  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] A ) 
C_  U. z ) )
3635rexbidv 2892 . . . 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
) )
37 icccmp.4 . . . 4  |-  S  =  { x  e.  ( A [,] B )  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }
3836, 37elrab2 3186 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
) )
397, 33, 38sylanbrc 677 . 2  |-  ( ph  ->  A  e.  S )
40 ssrab2 3500 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
4137, 40eqsstri 3448 . . . . 5  |-  S  C_  ( A [,] B )
4241sseli 3414 . . . 4  |-  ( y  e.  S  ->  y  e.  ( A [,] B
) )
43 elicc2 11724 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
441, 3, 43syl2anc 673 . . . . . 6  |-  ( ph  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
4544biimpa 492 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) )
4645simp3d 1044 . . . 4  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  <_  B )
4742, 46sylan2 482 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  y  <_  B )
4847ralrimiva 2809 . 2  |-  ( ph  ->  A. y  e.  S  y  <_  B )
4939, 48jca 541 1  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   {csn 3959   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   RRcr 9556   RR*cxr 9692    <_ cle 9694    - cmin 9880   (,)cioo 11660   [,]cicc 11663   abscabs 13374   ↾t crest 15397   topGenctg 15414
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-pre-lttri 9631  ax-pre-lttrn 9632
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-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-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-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-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-icc 11667
This theorem is referenced by:  icccmplem2  21919  icccmplem3  21920
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