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Theorem stoweidlem58 29699
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 2578 . . . 4  |-  F/ t  D  =  (/)
42, 3nfan 1859 . . 3  |-  F/ t ( ph  /\  D  =  (/) )
5 eqid 2433 . . 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 707 . . 3  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  (
t  e.  T  |->  a )  e.  A )
9 stoweidlem58.13 . . . 4  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
109adantr 462 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  B  e.  ( Clsd `  J )
)
11 stoweidlem58.17 . . . 4  |-  ( ph  ->  E  e.  RR+ )
1211adantr 462 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  E  e.  RR+ )
13 simpr 458 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  D  =  (/) )
141, 4, 5, 6, 8, 10, 12, 13stoweidlem18 29659 . 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 2569 . . . . 5  |-  F/_ t (/)
171, 16nfne 2693 . . . 4  |-  F/ t  D  =/=  (/)
182, 17nfan 1859 . . 3  |-  F/ t ( ph  /\  D  =/=  (/) )
19 eqid 2433 . . 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 2433 . . 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 462 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  J  e.  Comp )
26 stoweidlem58.8 . . . 4  |-  ( ph  ->  A  C_  C )
2726adantr 462 . . 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 1204 . . 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 1204 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  e.  A )
327adantlr 707 . . 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 707 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
359adantr 462 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  B  e.  ( Clsd `  J )
)
36 stoweidlem58.14 . . . 4  |-  ( ph  ->  D  e.  ( Clsd `  J ) )
3736adantr 462 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  e.  ( Clsd `  J )
)
38 stoweidlem58.15 . . . 4  |-  ( ph  ->  ( B  i^i  D
)  =  (/) )
3938adantr 462 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  ( B  i^i  D )  =  (/) )
40 simpr 458 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  =/=  (/) )
4111adantr 462 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  E  e.  RR+ )
42 stoweidlem58.18 . . . 4  |-  ( ph  ->  E  <  ( 1  /  3 ) )
4342adantr 462 . . 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 29698 . 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 2679 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    = wceq 1362   F/wnf 1592    e. wcel 1755   F/_wnfc 2556    =/= wne 2596   A.wral 2705   E.wrex 2706   {crab 2709    \ cdif 3313    i^i cin 3315    C_ wss 3316   (/)c0 3625   U.cuni 4079   class class class wbr 4280    e. cmpt 4338   ran crn 4828   ` cfv 5406  (class class class)co 6080   RRcr 9269   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   3c3 10360   RR+crp 10979   (,)cioo 11288   topGenctg 14359   Clsdccld 18462    Cn ccn 18670   Compccmp 18831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-cn 18673  df-cnp 18674  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-xms 19737  df-ms 19738  df-tms 19739
This theorem is referenced by:  stoweidlem59  29700
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