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.

Step	Hyp	Ref	Expression
1		stoweidlem58.1	. . 3 |- F/_ t D
2		stoweidlem58.3	. . . 4 |- F/ t ph
3	1	nfeq1 2631	. . . 4 |- F/ t D = (/)
4	2, 3	nfan 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 )
8	7	adantlr 714	. . 3 |- ( ( ( ph /\ D = (/) ) /\ a e. RR ) -> ( t e. T |-> a ) e. A )
9		stoweidlem58.13	. . . 4 |- ( ph -> B e. ( Clsd ` J ) )
10	9	adantr 465	. . 3 |- ( ( ph /\ D = (/) ) -> B e. ( Clsd ` J ) )
11		stoweidlem58.17	. . . 4 |- ( ph -> E e. RR+ )
12	11	adantr 465	. . 3 |- ( ( ph /\ D = (/) ) -> E e. RR+ )
13		simpr 461	. . 3 |- ( ( ph /\ D = (/) ) -> D = (/) )
14	1, 4, 5, 6, 8, 10, 12, 13	stoweidlem18 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 (/)
17	1, 16	nfne 2783	. . . 4 |- F/ t D =/= (/)
18	2, 17	nfan 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 )
25	24	adantr 465	. . 3 |- ( ( ph /\ D =/= (/) ) -> J e. Comp )
26		stoweidlem58.8	. . . 4 |- ( ph -> A C_ C )
27	26	adantr 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 )
29	28	3adant1r 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 )
31	30	3adant1r 1212	. . 3 |- ( ( ( ph /\ D =/= (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
32	7	adantlr 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 ) )
34	33	adantlr 714	. . 3 |- ( ( ( ph /\ D =/= (/) ) /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
35	9	adantr 465	. . 3 |- ( ( ph /\ D =/= (/) ) -> B e. ( Clsd ` J ) )
36		stoweidlem58.14	. . . 4 |- ( ph -> D e. ( Clsd ` J ) )
37	36	adantr 465	. . 3 |- ( ( ph /\ D =/= (/) ) -> D e. ( Clsd ` J ) )
38		stoweidlem58.15	. . . 4 |- ( ph -> ( B i^i D ) = (/) )
39	38	adantr 465	. . 3 |- ( ( ph /\ D =/= (/) ) -> ( B i^i D ) = (/) )
40		simpr 461	. . 3 |- ( ( ph /\ D =/= (/) ) -> D =/= (/) )
41	11	adantr 465	. . 3 |- ( ( ph /\ D =/= (/) ) -> E e. RR+ )
42		stoweidlem58.18	. . . 4 |- ( ph -> E < ( 1 / 3 ) )
43	42	adantr 465	. . 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 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 ) ) )
45	14, 44	pm2.61dane 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 ) ) )

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