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Theorem stoweidlem58 30024
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 2631 . . . 4  |-  F/ t  D  =  (/)
42, 3nfan 1866 . . 3  |-  F/ t ( ph  /\  D  =  (/) )
5 eqid 2454 . . 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 714 . . 3  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  (
t  e.  T  |->  a )  e.  A )
9 stoweidlem58.13 . . . 4  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
109adantr 465 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  B  e.  ( Clsd `  J )
)
11 stoweidlem58.17 . . . 4  |-  ( ph  ->  E  e.  RR+ )
1211adantr 465 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  E  e.  RR+ )
13 simpr 461 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  D  =  (/) )
141, 4, 5, 6, 8, 10, 12, 13stoweidlem18 29984 . 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 2616 . . . . 5  |-  F/_ t (/)
171, 16nfne 2783 . . . 4  |-  F/ t  D  =/=  (/)
182, 17nfan 1866 . . 3  |-  F/ t ( ph  /\  D  =/=  (/) )
19 eqid 2454 . . 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 2454 . . 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 465 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  J  e.  Comp )
26 stoweidlem58.8 . . . 4  |-  ( ph  ->  A  C_  C )
2726adantr 465 . . 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 1212 . . 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 1212 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  e.  A )
327adantlr 714 . . 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 714 . . 3  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
359adantr 465 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  B  e.  ( Clsd `  J )
)
36 stoweidlem58.14 . . . 4  |-  ( ph  ->  D  e.  ( Clsd `  J ) )
3736adantr 465 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  e.  ( Clsd `  J )
)
38 stoweidlem58.15 . . . 4  |-  ( ph  ->  ( B  i^i  D
)  =  (/) )
3938adantr 465 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  ( B  i^i  D )  =  (/) )
40 simpr 461 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  =/=  (/) )
4111adantr 465 . . 3  |-  ( (
ph  /\  D  =/=  (/) )  ->  E  e.  RR+ )
42 stoweidlem58.18 . . . 4  |-  ( ph  ->  E  <  ( 1  /  3 ) )
4342adantr 465 . . 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 30023 . 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 2770 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 965    = wceq 1370   F/wnf 1590    e. wcel 1758   F/_wnfc 2602    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    \ cdif 3436    i^i cin 3438    C_ wss 3439   (/)c0 3748   U.cuni 4202   class class class wbr 4403    |-> cmpt 4461   ran crn 4952   ` cfv 5529  (class class class)co 6203   RRcr 9396   0cc0 9397   1c1 9398    + caddc 9400    x. cmul 9402    < clt 9533    <_ cle 9534    - cmin 9710    / cdiv 10108   3c3 10487   RR+crp 11106   (,)cioo 11415   topGenctg 14499   Clsdccld 18762    Cn ccn 18970   Compccmp 19131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-mulf 9477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-1o 7033  df-2o 7034  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-fi 7776  df-sup 7806  df-oi 7839  df-card 8224  df-cda 8452  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-q 11069  df-rp 11107  df-xneg 11204  df-xadd 11205  df-xmul 11206  df-ioo 11419  df-ico 11421  df-icc 11422  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-rlim 13089  df-sum 13286  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-mulr 14375  df-starv 14376  df-sca 14377  df-vsca 14378  df-ip 14379  df-tset 14380  df-ple 14381  df-ds 14383  df-unif 14384  df-hom 14385  df-cco 14386  df-rest 14484  df-topn 14485  df-0g 14503  df-gsum 14504  df-topgen 14505  df-pt 14506  df-prds 14509  df-xrs 14563  df-qtop 14568  df-imas 14569  df-xps 14571  df-mre 14647  df-mrc 14648  df-acs 14650  df-mnd 15538  df-submnd 15588  df-mulg 15671  df-cntz 15958  df-cmn 16404  df-psmet 17944  df-xmet 17945  df-met 17946  df-bl 17947  df-mopn 17948  df-cnfld 17954  df-top 18645  df-bases 18647  df-topon 18648  df-topsp 18649  df-cld 18765  df-cn 18973  df-cnp 18974  df-cmp 19132  df-tx 19277  df-hmeo 19470  df-xms 20037  df-ms 20038  df-tms 20039
This theorem is referenced by:  stoweidlem59  30025
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