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Theorem icccmplem1 21453
Description: Lemma for icccmp 21456. (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 9660 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 icccmp.6 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 9660 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 icccmp.7 . . . 4  |-  ( ph  ->  A  <_  B )
6 lbicc2 11661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1228 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
8 icccmp.9 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
98, 7sseldd 3500 . . . . 5  |-  ( ph  ->  A  e.  U. U
)
10 eluni2 4255 . . . . 5  |-  ( A  e.  U. U  <->  E. u  e.  U  A  e.  u )
119, 10sylib 196 . . . 4  |-  ( ph  ->  E. u  e.  U  A  e.  u )
12 snssi 4176 . . . . . . . 8  |-  ( u  e.  U  ->  { u }  C_  U )
1312ad2antrl 727 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  C_  U )
14 snex 4697 . . . . . . . 8  |-  { u }  e.  _V
1514elpw 4021 . . . . . . 7  |-  ( { u }  e.  ~P U 
<->  { u }  C_  U )
1613, 15sylibr 212 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  ~P U )
17 snfi 7615 . . . . . . 7  |-  { u }  e.  Fin
1817a1i 11 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  Fin )
1916, 18elind 3684 . . . . 5  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  ( ~P U  i^i  Fin ) )
202adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  A  e.  RR* )
21 iccid 11599 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
2220, 21syl 16 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  =  { A } )
23 snssi 4176 . . . . . . 7  |-  ( A  e.  u  ->  { A }  C_  u )
2423ad2antll 728 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { A }  C_  u
)
2522, 24eqsstrd 3533 . . . . 5  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  C_  u )
26 unieq 4259 . . . . . . . 8  |-  ( z  =  { u }  ->  U. z  =  U. { u } )
27 vex 3112 . . . . . . . . 9  |-  u  e. 
_V
2827unisn 4266 . . . . . . . 8  |-  U. {
u }  =  u
2926, 28syl6eq 2514 . . . . . . 7  |-  ( z  =  { u }  ->  U. z  =  u )
3029sseq2d 3527 . . . . . 6  |-  ( z  =  { u }  ->  ( ( A [,] A )  C_  U. z  <->  ( A [,] A ) 
C_  u ) )
3130rspcev 3210 . . . . 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 661 . . . 4  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  E. z  e.  ( ~P U  i^i  Fin )
( A [,] A
)  C_  U. z
)
3311, 32rexlimddv 2953 . . 3  |-  ( ph  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
)
34 oveq2 6304 . . . . . 6  |-  ( x  =  A  ->  ( A [,] x )  =  ( A [,] A
) )
3534sseq1d 3526 . . . . 5  |-  ( x  =  A  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] A ) 
C_  U. z ) )
3635rexbidv 2968 . . . 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 3259 . . 3  |-  ( A  e.  S  <->  ( A  e.  ( A [,] B
)  /\  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
) )
397, 33, 38sylanbrc 664 . 2  |-  ( ph  ->  A  e.  S )
40 ssrab2 3581 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
4137, 40eqsstri 3529 . . . . 5  |-  S  C_  ( A [,] B )
4241sseli 3495 . . . 4  |-  ( y  e.  S  ->  y  e.  ( A [,] B
) )
43 elicc2 11614 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
441, 3, 43syl2anc 661 . . . . . 6  |-  ( ph  ->  ( y  e.  ( A [,] B )  <-> 
( y  e.  RR  /\  A  <_  y  /\  y  <_  B ) ) )
4544biimpa 484 . . . . 5  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  ( y  e.  RR  /\  A  <_ 
y  /\  y  <_  B ) )
4645simp3d 1010 . . . 4  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  <_  B )
4742, 46sylan2 474 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  y  <_  B )
4847ralrimiva 2871 . 2  |-  ( ph  ->  A. y  e.  S  y  <_  B )
4939, 48jca 532 1  |-  ( ph  ->  ( A  e.  S  /\  A. y  e.  S  y  <_  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3470    C_ wss 3471   ~Pcpw 4015   {csn 4032   U.cuni 4251   class class class wbr 4456    X. cxp 5006   ran crn 5009    |` cres 5010    o. ccom 5012   ` cfv 5594  (class class class)co 6296   Fincfn 7535   RRcr 9508   RR*cxr 9644    <_ cle 9646    - cmin 9824   (,)cioo 11554   [,]cicc 11557   abscabs 13079   ↾t crest 14838   topGenctg 14855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-icc 11561
This theorem is referenced by:  icccmplem2  21454  icccmplem3  21455
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