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Theorem stoweidlem58 27475
Description: This theorem proves Lemma 2 in [BrosowskiDeutsh] p. 91. Here D is used to represent the set A of Lemma 2, because here the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem58.1  |-  F/_ t D
stoweidlem58.2  |-  F/_ t U
stoweidlem58.3  |-  F/ t
ph
stoweidlem58.4  |-  K  =  ( topGen `  ran  (,) )
stoweidlem58.5  |-  T  = 
U. J
stoweidlem58.6  |-  C  =  ( J  Cn  K
)
stoweidlem58.7  |-  ( ph  ->  J  e.  Comp )
stoweidlem58.8  |-  ( ph  ->  A  C_  C )
stoweidlem58.9  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
stoweidlem58.10  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
stoweidlem58.11  |-  ( (
ph  /\  a  e.  RR )  ->  ( t  e.  T  |->  a )  e.  A )
stoweidlem58.12  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
stoweidlem58.13  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
stoweidlem58.14  |-  ( ph  ->  D  e.  ( Clsd `  J ) )
stoweidlem58.15  |-  ( ph  ->  ( B  i^i  D
)  =  (/) )
stoweidlem58.16  |-  U  =  ( T  \  B
)
stoweidlem58.17  |-  ( ph  ->  E  e.  RR+ )
stoweidlem58.18  |-  ( ph  ->  E  <  ( 1  /  3 ) )
Assertion
Ref Expression
stoweidlem58  |-  ( ph  ->  E. x  e.  A  ( A. t  e.  T  ( 0  <_  (
x `  t )  /\  ( x `  t
)  <_  1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E
)  <  ( x `  t ) ) )
Distinct variable groups:    f, a,
r, t, A, q    D, a, f, r    T, a, f, r, t    U, a, f, r    ph, a,
f, r    f, g,
r, t, A    f, E, g, r, t    x, f, g, t, A    B, f, g, r    f, J, g, r, t    g,
q, D    T, g    U, g    ph, g    D, q    T, q    U, q    ph, q    t, K    x, B    x, D    x, E    x, T
Allowed substitution hints:    ph( x, t)    B( t, q, a)    C( x, t, f, g, r, q, a)    D( t)    U( x, t)    E( q, a)    J( x, q, a)    K( x, f, g, r, q, a)

Proof of Theorem stoweidlem58
Dummy variables  e  h  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 stoweidlem58.1 . . 3  |-  F/_ t D
2 stoweidlem58.3 . . . 4  |-  F/ t
ph
31nfeq1 2532 . . . 4  |-  F/ t  D  =  (/)
42, 3nfan 1836 . . 3  |-  F/ t ( ph  /\  D  =  (/) )
5 eqid 2387 . . 3  |-  ( t  e.  T  |->  1 )  =  ( t  e.  T  |->  1 )
6 stoweidlem58.5 . . 3  |-  T  = 
U. J
7 stoweidlem58.11 . . . 4  |-  ( (
ph  /\  a  e.  RR )  ->  ( t  e.  T  |->  a )  e.  A )
87adantlr 696 . . 3  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  (
t  e.  T  |->  a )  e.  A )
9 stoweidlem58.13 . . . 4  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
109adantr 452 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  B  e.  ( Clsd `  J )
)
11 stoweidlem58.17 . . . 4  |-  ( ph  ->  E  e.  RR+ )
1211adantr 452 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  E  e.  RR+ )
13 simpr 448 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  D  =  (/) )
141, 4, 5, 6, 8, 10, 12, 13stoweidlem18 27435 . 2  |-  ( (
ph  /\  D  =  (/) )  ->  E. x  e.  A  ( A. t  e.  T  (
0  <_  ( x `  t )  /\  (
x `  t )  <_  1 )  /\  A. t  e.  D  (
x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  < 
( x `  t
) ) )
15 stoweidlem58.2 . . 3  |-  F/_ t U
16 nfcv 2523 . . . . 5  |-  F/_ t (/)
171, 16nfne 2641 . . . 4  |-  F/ t  D  =/=  (/)
182, 17nfan 1836 . . 3  |-  F/ t ( ph  /\  D  =/=  (/) )
19 eqid 2387 . . 3  |-  { h  e.  A  |  A. t  e.  T  (
0  <_  ( h `  t )  /\  (
h `  t )  <_  1 ) }  =  { h  e.  A  |  A. t  e.  T  ( 0  <_  (
h `  t )  /\  ( h `  t
)  <_  1 ) }
20 eqid 2387 . . 3  |-  { w  e.  J  |  A. e  e.  RR+  E. h  e.  A  ( A. t  e.  T  (
0  <_  ( h `  t )  /\  (
h `  t )  <_  1 )  /\  A. t  e.  w  (
h `  t )  <  e  /\  A. t  e.  ( T  \  U
) ( 1  -  e )  <  (
h `  t )
) }  =  {
w  e.  J  |  A. e  e.  RR+  E. h  e.  A  ( A. t  e.  T  (
0  <_  ( h `  t )  /\  (
h `  t )  <_  1 )  /\  A. t  e.  w  (
h `  t )  <  e  /\  A. t  e.  ( T  \  U
) ( 1  -  e )  <  (
h `  t )
) }
21 stoweidlem58.4 . . 3  |-  K  =  ( topGen `  ran  (,) )
22 stoweidlem58.6 . . 3  |-  C  =  ( J  Cn  K
)
23 stoweidlem58.16 . . 3  |-  U  =  ( T  \  B
)
24 stoweidlem58.7 . . . 4  |-  ( ph  ->  J  e.  Comp )
2524adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  J  e.  Comp )
26 stoweidlem58.8 . . . 4  |-  ( ph  ->  A  C_  C )
2726adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  A  C_  C
)
28 stoweidlem58.9 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
29283adant1r 1177 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  e.  A )
30 stoweidlem58.10 . . . 4  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
31303adant1r 1177 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  e.  A )
327adantlr 696 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  a  e.  RR )  ->  (
t  e.  T  |->  a )  e.  A )
33 stoweidlem58.12 . . . 4  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
3433adantlr 696 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
359adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  B  e.  ( Clsd `  J )
)
36 stoweidlem58.14 . . . 4  |-  ( ph  ->  D  e.  ( Clsd `  J ) )
3736adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  e.  ( Clsd `  J )
)
38 stoweidlem58.15 . . . 4  |-  ( ph  ->  ( B  i^i  D
)  =  (/) )
3938adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  ( B  i^i  D )  =  (/) )
40 simpr 448 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  =/=  (/) )
4111adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  E  e.  RR+ )
42 stoweidlem58.18 . . . 4  |-  ( ph  ->  E  <  ( 1  /  3 ) )
4342adantr 452 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  E  <  ( 1  /  3 ) )
441, 15, 18, 19, 20, 21, 6, 22, 23, 25, 27, 29, 31, 32, 34, 35, 37, 39, 40, 41, 43stoweidlem57 27474 . 2  |-  ( (
ph  /\  D  =/=  (/) )  ->  E. x  e.  A  ( A. t  e.  T  (
0  <_  ( x `  t )  /\  (
x `  t )  <_  1 )  /\  A. t  e.  D  (
x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E )  < 
( x `  t
) ) )
4514, 44pm2.61dane 2628 1  |-  ( ph  ->  E. x  e.  A  ( A. t  e.  T  ( 0  <_  (
x `  t )  /\  ( x `  t
)  <_  1 )  /\  A. t  e.  D  ( x `  t )  <  E  /\  A. t  e.  B  ( 1  -  E
)  <  ( x `  t ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   F/wnf 1550    = wceq 1649    e. wcel 1717   F/_wnfc 2510    =/= wne 2550   A.wral 2649   E.wrex 2650   {crab 2653    \ cdif 3260    i^i cin 3262    C_ wss 3263   (/)c0 3571   U.cuni 3957   class class class wbr 4153    e. cmpt 4207   ran crn 4819   ` cfv 5394  (class class class)co 6020   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    <_ cle 9054    - cmin 9223    / cdiv 9609   3c3 9982   RR+crp 10544   (,)cioo 10848   topGenctg 13592   Clsdccld 17003    Cn ccn 17210   Compccmp 17371
This theorem is referenced by:  stoweidlem59  27476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-fi 7351  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-q 10507  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-ioo 10852  df-ico 10854  df-icc 10855  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-rest 13577  df-topn 13578  df-topgen 13594  df-pt 13595  df-prds 13598  df-xrs 13653  df-0g 13654  df-gsum 13655  df-qtop 13660  df-imas 13661  df-xps 13663  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-submnd 14666  df-mulg 14742  df-cntz 15043  df-cmn 15341  df-xmet 16619  df-met 16620  df-bl 16621  df-mopn 16622  df-cnfld 16627  df-top 16886  df-bases 16888  df-topon 16889  df-topsp 16890  df-cld 17006  df-cn 17213  df-cnp 17214  df-cmp 17372  df-tx 17515  df-hmeo 17708  df-xms 18259  df-ms 18260  df-tms 18261
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