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Theorem stoweidlem58 27910
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 2432 . . . . 5  |-  F/_ t (/)
41, 3nfeq 2439 . . . 4  |-  F/ t  D  =  (/)
52, 4nfan 1783 . . 3  |-  F/ t ( ph  /\  D  =  (/) )
6 eqid 2296 . . 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 27870 . 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 2461 . . . . . 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 2552 . . . . 5  |-  F/ t  D  =/=  (/)
262, 25nfan 1783 . . . 4  |-  F/ t ( ph  /\  D  =/=  (/) )
27 eqid 2296 . . . 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 2296 . . . 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 27909 . . 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 1534    = wceq 1632    e. wcel 1696   F/_wnfc 2419    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   ran crn 4706   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   3c3 9812   RR+crp 10370   (,)cioo 10672   topGenctg 13358   Clsdccld 16769    Cn ccn 16970   Compccmp 17129
This theorem is referenced by:  stoweidlem59  27911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903
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