Theorem stoweidlem58 27674

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 2549	. . . 4 |- F/ t D = (/)
4	2, 3	nfan 1842	. . 3 |- F/ t ( ph /\ D = (/) )
5		eqid 2404	. . 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 696	. . 3 |- ( ( ( ph /\ D = (/) ) /\ a e. RR ) -> ( t e. T |-> a ) e. A )
9		stoweidlem58.13	. . . 4 |- ( ph -> B e. ( Clsd ` J ) )
10	9	adantr 452	. . 3 |- ( ( ph /\ D = (/) ) -> B e. ( Clsd ` J ) )
11		stoweidlem58.17	. . . 4 |- ( ph -> E e. RR+ )
12	11	adantr 452	. . 3 |- ( ( ph /\ D = (/) ) -> E e. RR+ )
13		simpr 448	. . 3 |- ( ( ph /\ D = (/) ) -> D = (/) )
14	1, 4, 5, 6, 8, 10, 12, 13	stoweidlem18 27634	. 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 2540	. . . . 5 |- F/_ t (/)
17	1, 16	nfne 2658	. . . 4 |- F/ t D =/= (/)
18	2, 17	nfan 1842	. . 3 |- F/ t ( ph /\ D =/= (/) )
19		eqid 2404	. . 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 2404	. . 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 452	. . 3 |- ( ( ph /\ D =/= (/) ) -> J e. Comp )
26		stoweidlem58.8	. . . 4 |- ( ph -> A C_ C )
27	26	adantr 452	. . 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 1177	. . 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 1177	. . 3 |- ( ( ( ph /\ D =/= (/) ) /\ f e. A /\ g e. A ) -> ( t e. T |-> ( ( f ` t ) x. ( g ` t ) ) ) e. A )
32	7	adantlr 696	. . 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 696	. . 3 |- ( ( ( ph /\ D =/= (/) ) /\ ( r e. T /\ t e. T /\ r =/= t ) ) -> E. q e. A ( q ` r ) =/= ( q ` t ) )
35	9	adantr 452	. . 3 |- ( ( ph /\ D =/= (/) ) -> B e. ( Clsd ` J ) )
36		stoweidlem58.14	. . . 4 |- ( ph -> D e. ( Clsd ` J ) )
37	36	adantr 452	. . 3 |- ( ( ph /\ D =/= (/) ) -> D e. ( Clsd ` J ) )
38		stoweidlem58.15	. . . 4 |- ( ph -> ( B i^i D ) = (/) )
39	38	adantr 452	. . 3 |- ( ( ph /\ D =/= (/) ) -> ( B i^i D ) = (/) )
40		simpr 448	. . 3 |- ( ( ph /\ D =/= (/) ) -> D =/= (/) )
41	11	adantr 452	. . 3 |- ( ( ph /\ D =/= (/) ) -> E e. RR+ )
42		stoweidlem58.18	. . . 4 |- ( ph -> E < ( 1 / 3 ) )
43	42	adantr 452	. . 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 27673	. 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 2645	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 359   /\ w3a 936   F/wnf 1550   = wceq 1649   e. wcel 1721   F/_wnfc 2527   =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670   \ cdif 3277   i^i cin 3279   C_ wss 3280   (/)c0 3588   U.cuni 3975   class class class wbr 4172   e. cmpt 4226   ran crn 4838   ` cfv 5413  (class class class)co 6040   RRcr 8945   0cc0 8946   1c1 8947   + caddc 8949   x. cmul 8951   < clt 9076   <_ cle 9077   - cmin 9247   / cdiv 9633   3c3 10006   RR+crp 10568   (,)cioo 10872   topGenctg 13620   Clsdccld 17035   Cn ccn 17242   Compccmp 17403

This theorem is referenced by:  stoweidlem59  27675

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-mulf 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-cn 17245  df-cnp 17246  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-xms 18303  df-ms 18304  df-tms 18305