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Theorem icccmplem1 20404
Description: Lemma for icccmp 20407. (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 9438 . . . 4  |-  ( ph  ->  A  e.  RR* )
3 icccmp.6 . . . . 5  |-  ( ph  ->  B  e.  RR )
43rexrd 9438 . . . 4  |-  ( ph  ->  B  e.  RR* )
5 icccmp.7 . . . 4  |-  ( ph  ->  A  <_  B )
6 lbicc2 11406 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
72, 4, 5, 6syl3anc 1218 . . 3  |-  ( ph  ->  A  e.  ( A [,] B ) )
8 icccmp.9 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  U. U )
98, 7sseldd 3362 . . . . 5  |-  ( ph  ->  A  e.  U. U
)
10 eluni2 4100 . . . . 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 4022 . . . . . . . 8  |-  ( u  e.  U  ->  { u }  C_  U )
1312ad2antrl 727 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  C_  U )
14 snex 4538 . . . . . . . 8  |-  { u }  e.  _V
1514elpw 3871 . . . . . . 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 7395 . . . . . . 7  |-  { u }  e.  Fin
1817a1i 11 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { u }  e.  Fin )
1916, 18elind 3545 . . . . 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 11350 . . . . . . 7  |-  ( A  e.  RR*  ->  ( A [,] A )  =  { A } )
2220, 21syl 16 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  =  { A } )
23 snssi 4022 . . . . . . 7  |-  ( A  e.  u  ->  { A }  C_  u )
2423ad2antll 728 . . . . . 6  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  ->  { A }  C_  u
)
2522, 24eqsstrd 3395 . . . . 5  |-  ( (
ph  /\  ( u  e.  U  /\  A  e.  u ) )  -> 
( A [,] A
)  C_  u )
26 unieq 4104 . . . . . . . 8  |-  ( z  =  { u }  ->  U. z  =  U. { u } )
27 vex 2980 . . . . . . . . 9  |-  u  e. 
_V
2827unisn 4111 . . . . . . . 8  |-  U. {
u }  =  u
2926, 28syl6eq 2491 . . . . . . 7  |-  ( z  =  { u }  ->  U. z  =  u )
3029sseq2d 3389 . . . . . 6  |-  ( z  =  { u }  ->  ( ( A [,] A )  C_  U. z  <->  ( A [,] A ) 
C_  u ) )
3130rspcev 3078 . . . . 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 2850 . . 3  |-  ( ph  ->  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] A )  C_  U. z
)
34 oveq2 6104 . . . . . 6  |-  ( x  =  A  ->  ( A [,] x )  =  ( A [,] A
) )
3534sseq1d 3388 . . . . 5  |-  ( x  =  A  ->  (
( A [,] x
)  C_  U. z  <->  ( A [,] A ) 
C_  U. z ) )
3635rexbidv 2741 . . . 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 3124 . . 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 3442 . . . . . 6  |-  { x  e.  ( A [,] B
)  |  E. z  e.  ( ~P U  i^i  Fin ) ( A [,] x )  C_  U. z }  C_  ( A [,] B )
4137, 40eqsstri 3391 . . . . 5  |-  S  C_  ( A [,] B )
4241sseli 3357 . . . 4  |-  ( y  e.  S  ->  y  e.  ( A [,] B
) )
43 elicc2 11365 . . . . . . 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 1002 . . . 4  |-  ( (
ph  /\  y  e.  ( A [,] B ) )  ->  y  <_  B )
4742, 46sylan2 474 . . 3  |-  ( (
ph  /\  y  e.  S )  ->  y  <_  B )
4847ralrimiva 2804 . 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 965    = wceq 1369    e. wcel 1756   A.wral 2720   E.wrex 2721   {crab 2724    i^i cin 3332    C_ wss 3333   ~Pcpw 3865   {csn 3882   U.cuni 4096   class class class wbr 4297    X. cxp 4843   ran crn 4846    |` cres 4847    o. ccom 4849   ` cfv 5423  (class class class)co 6096   Fincfn 7315   RRcr 9286   RR*cxr 9422    <_ cle 9424    - cmin 9600   (,)cioo 11305   [,]cicc 11308   abscabs 12728   ↾t crest 14364   topGenctg 14381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-pre-lttri 9361  ax-pre-lttrn 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-icc 11312
This theorem is referenced by:  icccmplem2  20405  icccmplem3  20406
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