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.

Step	Hyp	Ref	Expression
1		stoweidlem58.1	. . 3 |- F/_ t D
2		stoweidlem58.3	. . . 4 |- F/ t ph
3	1	nfeq1 2578	. . . 4 |- F/ t D = (/)
4	2, 3	nfan 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 )
8	7	adantlr 707	. . 3 |- ( ( ( ph /\ D = (/) ) /\ a e. RR ) -> ( t e. T |-> a ) e. A )
9		stoweidlem58.13	. . . 4 |- ( ph -> B e. ( Clsd ` J ) )
10	9	adantr 462	. . 3 |- ( ( ph /\ D = (/) ) -> B e. ( Clsd ` J ) )
11		stoweidlem58.17	. . . 4 |- ( ph -> E e. RR+ )
12	11	adantr 462	. . 3 |- ( ( ph /\ D = (/) ) -> E e. RR+ )
13		simpr 458	. . 3 |- ( ( ph /\ D = (/) ) -> D = (/) )
14	1, 4, 5, 6, 8, 10, 12, 13	stoweidlem18 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 (/)
17	1, 16	nfne 2693	. . . 4 |- F/ t D =/= (/)
18	2, 17	nfan 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 )
25	24	adantr 462	. . 3 |- ( ( ph /\ D =/= (/) ) -> J e. Comp )
26		stoweidlem58.8	. . . 4 |- ( ph -> A C_ C )
27	26	adantr 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 )
29	28	3adant1r 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 )
31	30	3adant1r 1204	. . 3 |- ( ( ( ph /\ D =/= (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
32	7	adantlr 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 ) )
34	33	adantlr 707	. . 3 |- ( ( ( ph /\ D =/= (/) ) /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
35	9	adantr 462	. . 3 |- ( ( ph /\ D =/= (/) ) -> B e. ( Clsd ` J ) )
36		stoweidlem58.14	. . . 4 |- ( ph -> D e. ( Clsd ` J ) )
37	36	adantr 462	. . 3 |- ( ( ph /\ D =/= (/) ) -> D e. ( Clsd ` J ) )
38		stoweidlem58.15	. . . 4 |- ( ph -> ( B i^i D ) = (/) )
39	38	adantr 462	. . 3 |- ( ( ph /\ D =/= (/) ) -> ( B i^i D ) = (/) )
40		simpr 458	. . 3 |- ( ( ph /\ D =/= (/) ) -> D =/= (/) )
41	11	adantr 462	. . 3 |- ( ( ph /\ D =/= (/) ) -> E e. RR+ )
42		stoweidlem58.18	. . . 4 |- ( ph -> E < ( 1 / 3 ) )
43	42	adantr 462	. . 3 |- ( ( ph /\ D =/= (/) ) -> E < ( 1 / 3 ) )
44	1, 15, 18, 19, 20, 21, 6, 22, 23, 25, 27, 29, 31, 32, 34, 35, 37, 39, 40, 41, 43	stoweidlem57 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 ) ) )
45	14, 44	pm2.61dane 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 ) ) )

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