stoweidlem38 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem38		Structured version Visualization version GIF version

Theorem stoweidlem38 42330

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_(t₀) = 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 ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
stoweidlem38.2	⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
stoweidlem38.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem38.4	⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄)
stoweidlem38.5	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)

**Assertion**
Ref	Expression
stoweidlem38	⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (0 ≤ (𝑃‘𝑆) ∧ (𝑃‘𝑆) ≤ 1))

Distinct variable groups: 𝑓,𝑖,𝑇 𝐴,𝑓 𝑓,𝐺 𝜑,𝑓,𝑖 ℎ,𝑖,𝑡,𝑇 𝐴,ℎ ℎ,𝐺,𝑡 ℎ,𝑍 𝑖,𝑀,𝑡 𝑆,𝑖 Allowed substitution hints: 𝜑(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 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	nnrecred 11691	. . . . 5 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
3	2	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (1 / 𝑀) ∈ ℝ)
4		fzfid 13344	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (1...𝑀) ∈ Fin)
5		stoweidlem38.1	. . . . . . . 8 ⊢ 𝑄 = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
6		stoweidlem38.4	. . . . . . . 8 ⊢ (𝜑 → 𝐺:(1...𝑀)⟶𝑄)
7		stoweidlem38.5	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
8	5, 6, 7	stoweidlem15 42307	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → (((𝐺‘𝑖)‘𝑆) ∈ ℝ ∧ 0 ≤ ((𝐺‘𝑖)‘𝑆) ∧ ((𝐺‘𝑖)‘𝑆) ≤ 1))
9	8	simp1d 1138	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑆) ∈ ℝ)
10	9	an32s 650	. . . . 5 ⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑆) ∈ ℝ)
11	4, 10	fsumrecl 15093	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ∈ ℝ)
12		1red 10644	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
13		0le1 11165	. . . . . . 7 ⊢ 0 ≤ 1
14	13	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ≤ 1)
15	1	nnred 11655	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
16	1	nngt0d 11689	. . . . . 6 ⊢ (𝜑 → 0 < 𝑀)
17		divge0 11511	. . . . . 6 ⊢ (((1 ∈ ℝ ∧ 0 ≤ 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀)) → 0 ≤ (1 / 𝑀))
18	12, 14, 15, 16, 17	syl22anc 836	. . . . 5 ⊢ (𝜑 → 0 ≤ (1 / 𝑀))
19	18	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ (1 / 𝑀))
20	8	simp2d 1139	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → 0 ≤ ((𝐺‘𝑖)‘𝑆))
21	20	an32s 650	. . . . 5 ⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 0 ≤ ((𝐺‘𝑖)‘𝑆))
22	4, 10, 21	fsumge0 15152	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆))
23	3, 11, 19, 22	mulge0d 11219	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)))
24		stoweidlem38.2	. . . 4 ⊢ 𝑃 = (𝑡 ∈ 𝑇 ↦ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑡)))
25	5, 24, 1, 6, 7	stoweidlem30 42322	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) = ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)))
26	23, 25	breqtrrd 5096	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 ≤ (𝑃‘𝑆))
27		1red 10644	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → 1 ∈ ℝ)
28	8	simp3d 1140	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑆 ∈ 𝑇) → ((𝐺‘𝑖)‘𝑆) ≤ 1)
29	28	an32s 650	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ∈ 𝑇) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐺‘𝑖)‘𝑆) ≤ 1)
30	4, 10, 27, 29	fsumle 15156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ Σ𝑖 ∈ (1...𝑀)1)
31		fzfid 13344	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
32		ax-1cn 10597	. . . . . . . . 9 ⊢ 1 ∈ ℂ
33		fsumconst 15147	. . . . . . . . 9 ⊢ (((1...𝑀) ∈ Fin ∧ 1 ∈ ℂ) → Σ𝑖 ∈ (1...𝑀)1 = ((♯‘(1...𝑀)) · 1))
34	31, 32, 33	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)1 = ((♯‘(1...𝑀)) · 1))
35	1	nnnn0d 11958	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
36		hashfz1 13709	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
37	35, 36	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
38	37	oveq1d 7173	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(1...𝑀)) · 1) = (𝑀 · 1))
39	1	nncnd 11656	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
40	39	mulid1d 10660	. . . . . . . 8 ⊢ (𝜑 → (𝑀 · 1) = 𝑀)
41	34, 38, 40	3eqtrd 2862	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (1...𝑀)1 = 𝑀)
42	41	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)1 = 𝑀)
43	30, 42	breqtrd 5094	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ 𝑀)
44	15	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 𝑀 ∈ ℝ)
45		1red 10644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 1 ∈ ℝ)
46		0lt1 11164	. . . . . . . 8 ⊢ 0 < 1
47	46	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 < 1)
48	15, 16	jca 514	. . . . . . . 8 ⊢ (𝜑 → (𝑀 ∈ ℝ ∧ 0 < 𝑀))
49	48	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑀 ∈ ℝ ∧ 0 < 𝑀))
50		divgt0 11510	. . . . . . 7 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀)) → 0 < (1 / 𝑀))
51	45, 47, 49, 50	syl21anc 835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → 0 < (1 / 𝑀))
52		lemul2 11495	. . . . . 6 ⊢ ((Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ((1 / 𝑀) ∈ ℝ ∧ 0 < (1 / 𝑀))) → (Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ 𝑀 ↔ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) ≤ ((1 / 𝑀) · 𝑀)))
53	11, 44, 3, 51, 52	syl112anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆) ≤ 𝑀 ↔ ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) ≤ ((1 / 𝑀) · 𝑀)))
54	43, 53	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → ((1 / 𝑀) · Σ𝑖 ∈ (1...𝑀)((𝐺‘𝑖)‘𝑆)) ≤ ((1 / 𝑀) · 𝑀))
55	25, 54	eqbrtrd 5090	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) ≤ ((1 / 𝑀) · 𝑀))
56	32	a1i 11	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℂ)
57	1	nnne0d 11690	. . . . . 6 ⊢ (𝜑 → 𝑀 ≠ 0)
58	56, 39, 57	3jca 1124	. . . . 5 ⊢ (𝜑 → (1 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑀 ≠ 0))
59	58	adantr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (1 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑀 ≠ 0))
60		divcan1 11309	. . . 4 ⊢ ((1 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑀 ≠ 0) → ((1 / 𝑀) · 𝑀) = 1)
61	59, 60	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → ((1 / 𝑀) · 𝑀) = 1)
62	55, 61	breqtrd 5094	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (𝑃‘𝑆) ≤ 1)
63	26, 62	jca 514	1 ⊢ ((𝜑 ∧ 𝑆 ∈ 𝑇) → (0 ≤ (𝑃‘𝑆) ∧ (𝑃‘𝑆) ≤ 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 {crab 3144 class class class wbr 5068 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 · cmul 10544 < clt 10677 ≤ cle 10678 / cdiv 11299 ℕcn 11640 ℕ₀cn0 11900 ...cfz 12895 ♯chash 13693 Σcsu 15044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045

This theorem is referenced by: stoweidlem44 42336

W3C validator