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Theorem stoweidlem58 27807
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
3 nfcv 2419 . . . . 5  |-  F/_ t (/)
41, 3nfeq 2426 . . . 4  |-  F/ t  D  =  (/)
52, 4nfan 1771 . . 3  |-  F/ t ( ph  /\  D  =  (/) )
6 eqid 2283 . . 3  |-  ( t  e.  T  |->  1 )  =  ( t  e.  T  |->  1 )
7 stoweidlem58.5 . . 3  |-  T  = 
U. J
8 simpll 730 . . . . 5  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  ph )
9 simpr 447 . . . . 5  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  a  e.  RR )
108, 9jca 518 . . . 4  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  ( ph  /\  a  e.  RR ) )
11 stoweidlem58.11 . . . 4  |-  ( (
ph  /\  a  e.  RR )  ->  ( t  e.  T  |->  a )  e.  A )
1210, 11syl 15 . . 3  |-  ( ( ( ph  /\  D  =  (/) )  /\  a  e.  RR )  ->  (
t  e.  T  |->  a )  e.  A )
13 stoweidlem58.13 . . . 4  |-  ( ph  ->  B  e.  ( Clsd `  J ) )
1413adantr 451 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  B  e.  ( Clsd `  J )
)
15 stoweidlem58.17 . . . 4  |-  ( ph  ->  E  e.  RR+ )
1615adantr 451 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  E  e.  RR+ )
17 simpr 447 . . 3  |-  ( (
ph  /\  D  =  (/) )  ->  D  =  (/) )
181, 5, 6, 7, 12, 14, 16, 17stoweidlem18 27767 . 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
) ) )
19 simpl 443 . . . 4  |-  ( (
ph  /\  -.  D  =  (/) )  ->  ph )
20 df-ne 2448 . . . . . 6  |-  ( D  =/=  (/)  <->  -.  D  =  (/) )
2120biimpri 197 . . . . 5  |-  ( -.  D  =  (/)  ->  D  =/=  (/) )
2221adantl 452 . . . 4  |-  ( (
ph  /\  -.  D  =  (/) )  ->  D  =/=  (/) )
2319, 22jca 518 . . 3  |-  ( (
ph  /\  -.  D  =  (/) )  ->  ( ph  /\  D  =/=  (/) ) )
24 stoweidlem58.2 . . . 4  |-  F/_ t U
251, 3nfne 2539 . . . . 5  |-  F/ t  D  =/=  (/)
262, 25nfan 1771 . . . 4  |-  F/ t ( ph  /\  D  =/=  (/) )
27 eqid 2283 . . . 4  |-  { 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 ) }
28 eqid 2283 . . . 4  |-  { 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 )
) }
29 stoweidlem58.4 . . . 4  |-  K  =  ( topGen `  ran  (,) )
30 stoweidlem58.6 . . . 4  |-  C  =  ( J  Cn  K
)
31 stoweidlem58.16 . . . 4  |-  U  =  ( T  \  B
)
32 stoweidlem58.7 . . . . 5  |-  ( ph  ->  J  e.  Comp )
3332adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  J  e.  Comp )
34 stoweidlem58.8 . . . . 5  |-  ( ph  ->  A  C_  C )
3534adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  A  C_  C
)
36 simp1l 979 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ph )
37 simp2 956 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  f  e.  A )
38 simp3 957 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  g  e.  A )
3936, 37, 383jca 1132 . . . . 5  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  ( ph  /\  f  e.  A  /\  g  e.  A
) )
40 stoweidlem58.9 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  +  ( g `  t ) ) )  e.  A )
4139, 40syl 15 . . . 4  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  +  ( g `
 t ) ) )  e.  A )
42 stoweidlem58.10 . . . . 5  |-  ( (
ph  /\  f  e.  A  /\  g  e.  A
)  ->  ( t  e.  T  |->  ( ( f `  t )  x.  ( g `  t ) ) )  e.  A )
4339, 42syl 15 . . . 4  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  f  e.  A  /\  g  e.  A )  ->  (
t  e.  T  |->  ( ( f `  t
)  x.  ( g `
 t ) ) )  e.  A )
44 simpll 730 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  a  e.  RR )  ->  ph )
45 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  a  e.  RR )  ->  a  e.  RR )
4644, 45jca 518 . . . . 5  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  a  e.  RR )  ->  ( ph  /\  a  e.  RR ) )
4746, 11syl 15 . . . 4  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  a  e.  RR )  ->  (
t  e.  T  |->  a )  e.  A )
48 simpll 730 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  ph )
49 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )
5048, 49jca 518 . . . . 5  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  ( ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t
) ) )
51 stoweidlem58.12 . . . . 5  |-  ( (
ph  /\  ( r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
5250, 51syl 15 . . . 4  |-  ( ( ( ph  /\  D  =/=  (/) )  /\  (
r  e.  T  /\  t  e.  T  /\  r  =/=  t ) )  ->  E. q  e.  A  ( q `  r
)  =/=  ( q `
 t ) )
5313adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  B  e.  ( Clsd `  J )
)
54 stoweidlem58.14 . . . . 5  |-  ( ph  ->  D  e.  ( Clsd `  J ) )
5554adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  e.  ( Clsd `  J )
)
56 stoweidlem58.15 . . . . 5  |-  ( ph  ->  ( B  i^i  D
)  =  (/) )
5756adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  ( B  i^i  D )  =  (/) )
58 simpr 447 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  D  =/=  (/) )
5915adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  E  e.  RR+ )
60 stoweidlem58.18 . . . . 5  |-  ( ph  ->  E  <  ( 1  /  3 ) )
6160adantr 451 . . . 4  |-  ( (
ph  /\  D  =/=  (/) )  ->  E  <  ( 1  /  3 ) )
621, 24, 26, 27, 28, 29, 7, 30, 31, 33, 35, 41, 43, 47, 52, 53, 55, 57, 58, 59, 61stoweidlem57 27806 . . 3  |-  ( (
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
) ) )
6323, 62syl 15 . 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
) ) )
6418, 63pm2.61dan 766 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:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934   F/wnf 1531    = wceq 1623    e. wcel 1684   F/_wnfc 2406    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   3c3 9796   RR+crp 10354   (,)cioo 10656   topGenctg 13342   Clsdccld 16753    Cn ccn 16954   Compccmp 17113
This theorem is referenced by:  stoweidlem59  27808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-cn 16957  df-cnp 16958  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887
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