Theorem stoweidlem38 40255

Description: This lemma is used to prove the existence of a function p as in Lemma 1 of [BrosowskiDeutsh] p. 90: p is in the subalgebra, such that 0 <= p <= 1, p_(t_0) = 0, and p > 0 on T - U. Z is used for t₀, P is used for p, ( G ` i ) is used for p_(t_i). (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem38.1	|- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
stoweidlem38.2	|- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
stoweidlem38.3	|- ( ph -> M e. NN )
stoweidlem38.4	|- ( ph -> G : ( 1 ... M ) --> Q )
stoweidlem38.5	|- ( ( ph /\ f e. A ) -> f : T --> RR )

Assertion

Ref	Expression
stoweidlem38	|- ( ( ph /\ S e. T ) -> ( 0 <_ ( P ` S ) /\ ( P ` S ) <_ 1 ) )

Distinct variable groups:   f, i, T   A, f   f, G   ph, f, i   h, i, t, T   A, h   h, G, t   h, Z   i, M, t   S, i

Allowed substitution hints:   ph( t, h)   A( t, i)   P( t, f, h, i)   Q( t, f, h, i)   S( t, f, h)   G( i)   M( f, h)   Z( t, f, i)

Proof of Theorem stoweidlem38

Step	Hyp	Ref	Expression
1		stoweidlem38.3	. . . . . 6 |- ( ph -> M e. NN )
2	1	nnrecred 11066	. . . . 5 |- ( ph -> ( 1 / M ) e. RR )
3	2	adantr 481	. . . 4 |- ( ( ph /\ S e. T ) -> ( 1 / M ) e. RR )
4		fzfid 12772	. . . . 5 |- ( ( ph /\ S e. T ) -> ( 1 ... M ) e. Fin )
5		stoweidlem38.1	. . . . . . . 8 |- Q = { h e. A | ( ( h ` Z ) = 0 /\ A. t e. T ( 0 <_ ( h ` t ) /\ ( h ` t ) <_ 1 ) ) }
6		stoweidlem38.4	. . . . . . . 8 |- ( ph -> G : ( 1 ... M ) --> Q )
7		stoweidlem38.5	. . . . . . . 8 |- ( ( ph /\ f e. A ) -> f : T --> RR )
8	5, 6, 7	stoweidlem15 40232	. . . . . . 7 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> ( ( ( G ` i ) ` S ) e. RR /\ 0 <_ ( ( G ` i ) ` S ) /\ ( ( G ` i ) ` S ) <_ 1 ) )
9	8	simp1d 1073	. . . . . 6 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> ( ( G ` i ) ` S ) e. RR )
10	9	an32s 846	. . . . 5 |- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` S ) e. RR )
11	4, 10	fsumrecl 14465	. . . 4 |- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) e. RR )
12		1red 10055	. . . . . 6 |- ( ph -> 1 e. RR )
13		0le1 10551	. . . . . . 7 |- 0 <_ 1
14	13	a1i 11	. . . . . 6 |- ( ph -> 0 <_ 1 )
15	1	nnred 11035	. . . . . 6 |- ( ph -> M e. RR )
16	1	nngt0d 11064	. . . . . 6 |- ( ph -> 0 < M )
17		divge0 10892	. . . . . 6 |- ( ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( 1 / M ) )
18	12, 14, 15, 16, 17	syl22anc 1327	. . . . 5 |- ( ph -> 0 <_ ( 1 / M ) )
19	18	adantr 481	. . . 4 |- ( ( ph /\ S e. T ) -> 0 <_ ( 1 / M ) )
20	8	simp2d 1074	. . . . . 6 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> 0 <_ ( ( G ` i ) ` S ) )
21	20	an32s 846	. . . . 5 |- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> 0 <_ ( ( G ` i ) ` S ) )
22	4, 10, 21	fsumge0 14527	. . . 4 |- ( ( ph /\ S e. T ) -> 0 <_ sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) )
23	3, 11, 19, 22	mulge0d 10604	. . 3 |- ( ( ph /\ S e. T ) -> 0 <_ ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )
24		stoweidlem38.2	. . . 4 |- P = ( t e. T |-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` t ) ) )
25	5, 24, 1, 6, 7	stoweidlem30 40247	. . 3 |- ( ( ph /\ S e. T ) -> ( P ` S ) = ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) )
26	23, 25	breqtrrd 4681	. 2 |- ( ( ph /\ S e. T ) -> 0 <_ ( P ` S ) )
27		1red 10055	. . . . . . 7 |- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> 1 e. RR )
28	8	simp3d 1075	. . . . . . . 8 |- ( ( ( ph /\ i e. ( 1 ... M ) ) /\ S e. T ) -> ( ( G ` i ) ` S ) <_ 1 )
29	28	an32s 846	. . . . . . 7 |- ( ( ( ph /\ S e. T ) /\ i e. ( 1 ... M ) ) -> ( ( G ` i ) ` S ) <_ 1 )
30	4, 10, 27, 29	fsumle 14531	. . . . . 6 |- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ sum_ i e. ( 1 ... M ) 1 )
31		fzfid 12772	. . . . . . . . 9 |- ( ph -> ( 1 ... M ) e. Fin )
32		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
33		fsumconst 14522	. . . . . . . . 9 |- ( ( ( 1 ... M ) e. Fin /\ 1 e. CC ) -> sum_ i e. ( 1 ... M ) 1 = ( ( # ` ( 1 ... M ) ) x. 1 ) )
34	31, 32, 33	sylancl 694	. . . . . . . 8 |- ( ph -> sum_ i e. ( 1 ... M ) 1 = ( ( # ` ( 1 ... M ) ) x. 1 ) )
35	1	nnnn0d 11351	. . . . . . . . . 10 |- ( ph -> M e. NN0 )
36		hashfz1 13134	. . . . . . . . . 10 |- ( M e. NN0 -> ( # ` ( 1 ... M ) ) = M )
37	35, 36	syl 17	. . . . . . . . 9 |- ( ph -> ( # ` ( 1 ... M ) ) = M )
38	37	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( # ` ( 1 ... M ) ) x. 1 ) = ( M x. 1 ) )
39	1	nncnd 11036	. . . . . . . . 9 |- ( ph -> M e. CC )
40	39	mulid1d 10057	. . . . . . . 8 |- ( ph -> ( M x. 1 ) = M )
41	34, 38, 40	3eqtrd 2660	. . . . . . 7 |- ( ph -> sum_ i e. ( 1 ... M ) 1 = M )
42	41	adantr 481	. . . . . 6 |- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) 1 = M )
43	30, 42	breqtrd 4679	. . . . 5 |- ( ( ph /\ S e. T ) -> sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ M )
44	15	adantr 481	. . . . . 6 |- ( ( ph /\ S e. T ) -> M e. RR )
45		1red 10055	. . . . . . 7 |- ( ( ph /\ S e. T ) -> 1 e. RR )
46		0lt1 10550	. . . . . . . 8 |- 0 < 1
47	46	a1i 11	. . . . . . 7 |- ( ( ph /\ S e. T ) -> 0 < 1 )
48	15, 16	jca 554	. . . . . . . 8 |- ( ph -> ( M e. RR /\ 0 < M ) )
49	48	adantr 481	. . . . . . 7 |- ( ( ph /\ S e. T ) -> ( M e. RR /\ 0 < M ) )
50		divgt0 10891	. . . . . . 7 |- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( M e. RR /\ 0 < M ) ) -> 0 < ( 1 / M ) )
51	45, 47, 49, 50	syl21anc 1325	. . . . . 6 |- ( ( ph /\ S e. T ) -> 0 < ( 1 / M ) )
52		lemul2 10876	. . . . . 6 |- ( ( sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) e. RR /\ M e. RR /\ ( ( 1 / M ) e. RR /\ 0 < ( 1 / M ) ) ) -> ( sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ M <-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) <_ ( ( 1 / M ) x. M ) ) )
53	11, 44, 3, 51, 52	syl112anc 1330	. . . . 5 |- ( ( ph /\ S e. T ) -> ( sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) <_ M <-> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) <_ ( ( 1 / M ) x. M ) ) )
54	43, 53	mpbid 222	. . . 4 |- ( ( ph /\ S e. T ) -> ( ( 1 / M ) x. sum_ i e. ( 1 ... M ) ( ( G ` i ) ` S ) ) <_ ( ( 1 / M ) x. M ) )
55	25, 54	eqbrtrd 4675	. . 3 |- ( ( ph /\ S e. T ) -> ( P ` S ) <_ ( ( 1 / M ) x. M ) )
56	32	a1i 11	. . . . . 6 |- ( ph -> 1 e. CC )
57	1	nnne0d 11065	. . . . . 6 |- ( ph -> M =/= 0 )
58	56, 39, 57	3jca 1242	. . . . 5 |- ( ph -> ( 1 e. CC /\ M e. CC /\ M =/= 0 ) )
59	58	adantr 481	. . . 4 |- ( ( ph /\ S e. T ) -> ( 1 e. CC /\ M e. CC /\ M =/= 0 ) )
60		divcan1 10694	. . . 4 |- ( ( 1 e. CC /\ M e. CC /\ M =/= 0 ) -> ( ( 1 / M ) x. M ) = 1 )
61	59, 60	syl 17	. . 3 |- ( ( ph /\ S e. T ) -> ( ( 1 / M ) x. M ) = 1 )
62	55, 61	breqtrd 4679	. 2 |- ( ( ph /\ S e. T ) -> ( P ` S ) <_ 1 )
63	26, 62	jca 554	1 |- ( ( ph /\ S e. T ) -> ( 0 <_ ( P ` S ) /\ ( P ` S ) <_ 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   NN0cn0 11292   ...cfz 12326   #chash 13117   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  stoweidlem44  40261