stoweidlem11 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem11 43896

Description: This lemma is used to prove that there is a function 𝑔 as in the proof of [BrosowskiDeutsh] p. 92 (at the top of page 92): this lemma proves that g(t) < ( j + 1 / 3 ) * ε. Here 𝐸 is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem11.1	⊢ (𝜑 → 𝑁 ∈ ℕ)
stoweidlem11.2	⊢ (𝜑 → 𝑡 ∈ 𝑇)
stoweidlem11.3	⊢ (𝜑 → 𝑗 ∈ (1...𝑁))
stoweidlem11.4	⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ)
stoweidlem11.5	⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ≤ 1)
stoweidlem11.6	⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → ((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁))
stoweidlem11.7	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem11.8	⊢ (𝜑 → 𝐸 < (1 / 3))

**Assertion**
Ref	Expression
stoweidlem11	⊢ (𝜑 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸))

Distinct variable groups: 𝑖,𝑗 𝑡,𝑖,𝐸 𝑖,𝑁,𝑡 𝜑,𝑖 𝑡,𝑇 𝑡,𝑋 Allowed substitution hints: 𝜑(𝑡,𝑗) 𝑇(𝑖,𝑗) 𝐸(𝑗) 𝑁(𝑗) 𝑋(𝑖,𝑗)

**Proof of Theorem stoweidlem11**
Step	Hyp	Ref	Expression
1		stoweidlem11.2	. . 3 ⊢ (𝜑 → 𝑡 ∈ 𝑇)
2		sumex 15498	. . 3 ⊢ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ V
3		eqid 2736	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
4	3	fvmpt2 6942	. . 3 ⊢ ((𝑡 ∈ 𝑇 ∧ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ V) → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
5	1, 2, 4	sylancl 586	. 2 ⊢ (𝜑 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) = Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))
6		fzfid 13794	. . . 4 ⊢ (𝜑 → (0...𝑁) ∈ Fin)
7		stoweidlem11.7	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
8	7	rpred 12873	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝐸 ∈ ℝ)
10		stoweidlem11.4	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ)
11	1	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝑡 ∈ 𝑇)
12	10, 11	ffvelcdmd 7018	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
13	9, 12	remulcld 11106	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
14	6, 13	fsumrecl 15545	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
15		stoweidlem11.3	. . . . . . 7 ⊢ (𝜑 → 𝑗 ∈ (1...𝑁))
16	15	elfzelzd 13358	. . . . . 6 ⊢ (𝜑 → 𝑗 ∈ ℤ)
17	16	zred 12527	. . . . 5 ⊢ (𝜑 → 𝑗 ∈ ℝ)
18	8, 17	remulcld 11106	. . . 4 ⊢ (𝜑 → (𝐸 · 𝑗) ∈ ℝ)
19		stoweidlem11.1	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
20	19	nnred 12089	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℝ)
21	20, 17	resubcld 11504	. . . . . 6 ⊢ (𝜑 → (𝑁 − 𝑗) ∈ ℝ)
22		1red 11077	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
23	21, 22	readdcld 11105	. . . . 5 ⊢ (𝜑 → ((𝑁 − 𝑗) + 1) ∈ ℝ)
24	8, 19	nndivred 12128	. . . . . 6 ⊢ (𝜑 → (𝐸 / 𝑁) ∈ ℝ)
25	8, 24	remulcld 11106	. . . . 5 ⊢ (𝜑 → (𝐸 · (𝐸 / 𝑁)) ∈ ℝ)
26	23, 25	remulcld 11106	. . . 4 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))) ∈ ℝ)
27	18, 26	readdcld 11105	. . 3 ⊢ (𝜑 → ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ∈ ℝ)
28		3re 12154	. . . . . . 7 ⊢ 3 ∈ ℝ
29	28	a1i 11	. . . . . 6 ⊢ (𝜑 → 3 ∈ ℝ)
30		3ne0 12180	. . . . . . 7 ⊢ 3 ≠ 0
31	30	a1i 11	. . . . . 6 ⊢ (𝜑 → 3 ≠ 0)
32	29, 31	rereccld 11903	. . . . 5 ⊢ (𝜑 → (1 / 3) ∈ ℝ)
33	17, 32	readdcld 11105	. . . 4 ⊢ (𝜑 → (𝑗 + (1 / 3)) ∈ ℝ)
34	33, 8	remulcld 11106	. . 3 ⊢ (𝜑 → ((𝑗 + (1 / 3)) · 𝐸) ∈ ℝ)
35		fzfid 13794	. . . . . 6 ⊢ (𝜑 → (0...(𝑗 − 1)) ∈ Fin)
36	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 𝐸 ∈ ℝ)
37		elfzelz 13357	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ∈ ℤ)
38		peano2zm 12464	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℤ → (𝑗 − 1) ∈ ℤ)
39	15, 37, 38	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 − 1) ∈ ℤ)
40	19	nnzd 12526	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
41	17, 22	resubcld 11504	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 − 1) ∈ ℝ)
42	17	lem1d 12009	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑗 − 1) ≤ 𝑗)
43		elfzuz3 13354	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (1...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑗))
44		eluzle 12696	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ_≥‘𝑗) → 𝑗 ≤ 𝑁)
45	15, 43, 44	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑗 ≤ 𝑁)
46	41, 17, 20, 42, 45	letrd 11233	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 − 1) ≤ 𝑁)
47		eluz2 12689	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑗 − 1)) ↔ ((𝑗 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑗 − 1) ≤ 𝑁))
48	39, 40, 46, 47	syl3anbrc 1342	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑗 − 1)))
49		fzss2 13397	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑗 − 1)) → (0...(𝑗 − 1)) ⊆ (0...𝑁))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0...(𝑗 − 1)) ⊆ (0...𝑁))
51	50	sselda 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 𝑖 ∈ (0...𝑁))
52	51, 12	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
53	36, 52	remulcld 11106	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
54	35, 53	fsumrecl 15545	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
55	54, 26	readdcld 11105	. . . 4 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ∈ ℝ)
56	17	ltm1d 12008	. . . . . . 7 ⊢ (𝜑 → (𝑗 − 1) < 𝑗)
57		fzdisj 13384	. . . . . . 7 ⊢ ((𝑗 − 1) < 𝑗 → ((0...(𝑗 − 1)) ∩ (𝑗...𝑁)) = ∅)
58	56, 57	syl 17	. . . . . 6 ⊢ (𝜑 → ((0...(𝑗 − 1)) ∩ (𝑗...𝑁)) = ∅)
59		fzssp1 13400	. . . . . . . . . 10 ⊢ (0...(𝑁 − 1)) ⊆ (0...((𝑁 − 1) + 1))
60	19	nncnd 12090	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
61		1cnd 11071	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
62	60, 61	npcand 11437	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
63	62	oveq2d 7353	. . . . . . . . . 10 ⊢ (𝜑 → (0...((𝑁 − 1) + 1)) = (0...𝑁))
64	59, 63	sseqtrid 3984	. . . . . . . . 9 ⊢ (𝜑 → (0...(𝑁 − 1)) ⊆ (0...𝑁))
65		1zzd 12452	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℤ)
66		fzsubel 13393	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑗 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 − 1) ∈ ((1 − 1)...(𝑁 − 1))))
67	65, 40, 16, 65, 66	syl22anc 836	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 − 1) ∈ ((1 − 1)...(𝑁 − 1))))
68	15, 67	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 − 1) ∈ ((1 − 1)...(𝑁 − 1)))
69		1m1e0 12146	. . . . . . . . . . 11 ⊢ (1 − 1) = 0
70	69	oveq1i 7347	. . . . . . . . . 10 ⊢ ((1 − 1)...(𝑁 − 1)) = (0...(𝑁 − 1))
71	68, 70	eleqtrdi 2847	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 − 1) ∈ (0...(𝑁 − 1)))
72	64, 71	sseldd 3933	. . . . . . . 8 ⊢ (𝜑 → (𝑗 − 1) ∈ (0...𝑁))
73		fzsplit 13383	. . . . . . . 8 ⊢ ((𝑗 − 1) ∈ (0...𝑁) → (0...𝑁) = ((0...(𝑗 − 1)) ∪ (((𝑗 − 1) + 1)...𝑁)))
74	72, 73	syl 17	. . . . . . 7 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑗 − 1)) ∪ (((𝑗 − 1) + 1)...𝑁)))
75	16	zcnd 12528	. . . . . . . . . 10 ⊢ (𝜑 → 𝑗 ∈ ℂ)
76	75, 61	npcand 11437	. . . . . . . . 9 ⊢ (𝜑 → ((𝑗 − 1) + 1) = 𝑗)
77	76	oveq1d 7352	. . . . . . . 8 ⊢ (𝜑 → (((𝑗 − 1) + 1)...𝑁) = (𝑗...𝑁))
78	77	uneq2d 4110	. . . . . . 7 ⊢ (𝜑 → ((0...(𝑗 − 1)) ∪ (((𝑗 − 1) + 1)...𝑁)) = ((0...(𝑗 − 1)) ∪ (𝑗...𝑁)))
79	74, 78	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑗 − 1)) ∪ (𝑗...𝑁)))
80	7	rpcnd 12875	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℂ)
81	80	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → 𝐸 ∈ ℂ)
82	12	recnd 11104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ∈ ℂ)
83	81, 82	mulcld 11096	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℂ)
84	58, 79, 6, 83	fsumsplit 15552	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) = (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))))
85		fzfid 13794	. . . . . . 7 ⊢ (𝜑 → (𝑗...𝑁) ∈ Fin)
86	8	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝐸 ∈ ℝ)
87		0zd 12432	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ∈ ℤ)
88		0red 11079	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ ℝ)
89		0le1 11599	. . . . . . . . . . . . . . 15 ⊢ 0 ≤ 1
90	89	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ≤ 1)
91		elfzuz 13353	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1...𝑁) → 𝑗 ∈ (ℤ_≥‘1))
92	15, 91	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑗 ∈ (ℤ_≥‘1))
93		eluz2 12689	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤ 𝑗))
94	92, 93	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤ 𝑗))
95	94	simp3d 1143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ≤ 𝑗)
96	88, 22, 17, 90, 95	letrd 11233	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 ≤ 𝑗)
97		eluz2 12689	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ (ℤ_≥‘0) ↔ (0 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
98	87, 16, 96, 97	syl3anbrc 1342	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑗 ∈ (ℤ_≥‘0))
99		fzss1 13396	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ_≥‘0) → (𝑗...𝑁) ⊆ (0...𝑁))
100	98, 99	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗...𝑁) ⊆ (0...𝑁))
101	100	sselda 3932	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝑖 ∈ (0...𝑁))
102	101, 10	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝑋‘𝑖):𝑇⟶ℝ)
103	1	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝑡 ∈ 𝑇)
104	102, 103	ffvelcdmd 7018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → ((𝑋‘𝑖)‘𝑡) ∈ ℝ)
105	86, 104	remulcld 11106	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
106	85, 105	fsumrecl 15545	. . . . . 6 ⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) ∈ ℝ)
107		eluzfz2 13365	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ_≥‘𝑗) → 𝑁 ∈ (𝑗...𝑁))
108		ne0i 4281	. . . . . . . . 9 ⊢ (𝑁 ∈ (𝑗...𝑁) → (𝑗...𝑁) ≠ ∅)
109	15, 43, 107, 108	4syl 19	. . . . . . . 8 ⊢ (𝜑 → (𝑗...𝑁) ≠ ∅)
110	19	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 𝑁 ∈ ℕ)
111	86, 110	nndivred 12128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 / 𝑁) ∈ ℝ)
112	86, 111	remulcld 11106	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 · (𝐸 / 𝑁)) ∈ ℝ)
113		stoweidlem11.6	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → ((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁))
114	7	rpgt0d 12876	. . . . . . . . . . 11 ⊢ (𝜑 → 0 < 𝐸)
115	114	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → 0 < 𝐸)
116		ltmul2 11927	. . . . . . . . . 10 ⊢ ((((𝑋‘𝑖)‘𝑡) ∈ ℝ ∧ (𝐸 / 𝑁) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁) ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) < (𝐸 · (𝐸 / 𝑁))))
117	104, 111, 86, 115, 116	syl112anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (((𝑋‘𝑖)‘𝑡) < (𝐸 / 𝑁) ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) < (𝐸 · (𝐸 / 𝑁))))
118	113, 117	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑗...𝑁)) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) < (𝐸 · (𝐸 / 𝑁)))
119	85, 109, 105, 112, 118	fsumlt 15611	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)))
120	19	nnne0d 12124	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ≠ 0)
121	80, 60, 120	divcld 11852	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑁) ∈ ℂ)
122	80, 121	mulcld 11096	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 · (𝐸 / 𝑁)) ∈ ℂ)
123		fsumconst 15601	. . . . . . . . 9 ⊢ (((𝑗...𝑁) ∈ Fin ∧ (𝐸 · (𝐸 / 𝑁)) ∈ ℂ) → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)) = ((♯‘(𝑗...𝑁)) · (𝐸 · (𝐸 / 𝑁))))
124	85, 122, 123	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)) = ((♯‘(𝑗...𝑁)) · (𝐸 · (𝐸 / 𝑁))))
125		hashfz 14242	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑗) → (♯‘(𝑗...𝑁)) = ((𝑁 − 𝑗) + 1))
126	15, 43, 125	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(𝑗...𝑁)) = ((𝑁 − 𝑗) + 1))
127	126	oveq1d 7352	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝑗...𝑁)) · (𝐸 · (𝐸 / 𝑁))) = (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))
128	124, 127	eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · (𝐸 / 𝑁)) = (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))
129	119, 128	breqtrd 5118	. . . . . 6 ⊢ (𝜑 → Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))
130	106, 26, 54, 129	ltadd2dd 11235	. . . . 5 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + Σ𝑖 ∈ (𝑗...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡))) < (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))))
131	84, 130	eqbrtrd 5114	. . . 4 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))))
132		stoweidlem11.5	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑁)) → ((𝑋‘𝑖)‘𝑡) ≤ 1)
133	51, 132	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → ((𝑋‘𝑖)‘𝑡) ≤ 1)
134		1red 11077	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 1 ∈ ℝ)
135	114	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → 0 < 𝐸)
136		lemul2 11929	. . . . . . . . . 10 ⊢ ((((𝑋‘𝑖)‘𝑡) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (((𝑋‘𝑖)‘𝑡) ≤ 1 ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 1)))
137	52, 134, 36, 135, 136	syl112anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (((𝑋‘𝑖)‘𝑡) ≤ 1 ↔ (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 1)))
138	133, 137	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 1))
139	80	mulid1d 11093	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 · 1) = 𝐸)
140	139	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · 1) = 𝐸)
141	138, 140	breqtrd 5118	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...(𝑗 − 1))) → (𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ 𝐸)
142	35, 53, 36, 141	fsumle 15610	. . . . . 6 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ Σ𝑖 ∈ (0...(𝑗 − 1))𝐸)
143		fsumconst 15601	. . . . . . . 8 ⊢ (((0...(𝑗 − 1)) ∈ Fin ∧ 𝐸 ∈ ℂ) → Σ𝑖 ∈ (0...(𝑗 − 1))𝐸 = ((♯‘(0...(𝑗 − 1))) · 𝐸))
144	35, 80, 143	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))𝐸 = ((♯‘(0...(𝑗 − 1))) · 𝐸))
145		0z 12431	. . . . . . . . . . 11 ⊢ 0 ∈ ℤ
146		1e0p1 12580	. . . . . . . . . . . . 13 ⊢ 1 = (0 + 1)
147	146	fveq2i 6828	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘1) = (ℤ_≥‘(0 + 1))
148	92, 147	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑗 ∈ (ℤ_≥‘(0 + 1)))
149		eluzp1m1 12709	. . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑗 ∈ (ℤ_≥‘(0 + 1))) → (𝑗 − 1) ∈ (ℤ_≥‘0))
150	145, 148, 149	sylancr 587	. . . . . . . . . 10 ⊢ (𝜑 → (𝑗 − 1) ∈ (ℤ_≥‘0))
151		hashfz 14242	. . . . . . . . . 10 ⊢ ((𝑗 − 1) ∈ (ℤ_≥‘0) → (♯‘(0...(𝑗 − 1))) = (((𝑗 − 1) − 0) + 1))
152	150, 151	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(0...(𝑗 − 1))) = (((𝑗 − 1) − 0) + 1))
153	75, 61	subcld 11433	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 − 1) ∈ ℂ)
154	153	subid1d 11422	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑗 − 1) − 0) = (𝑗 − 1))
155	154	oveq1d 7352	. . . . . . . . 9 ⊢ (𝜑 → (((𝑗 − 1) − 0) + 1) = ((𝑗 − 1) + 1))
156	152, 155, 76	3eqtrd 2780	. . . . . . . 8 ⊢ (𝜑 → (♯‘(0...(𝑗 − 1))) = 𝑗)
157	156	oveq1d 7352	. . . . . . 7 ⊢ (𝜑 → ((♯‘(0...(𝑗 − 1))) · 𝐸) = (𝑗 · 𝐸))
158	75, 80	mulcomd 11097	. . . . . . 7 ⊢ (𝜑 → (𝑗 · 𝐸) = (𝐸 · 𝑗))
159	144, 157, 158	3eqtrd 2780	. . . . . 6 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))𝐸 = (𝐸 · 𝑗))
160	142, 159	breqtrd 5118	. . . . 5 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) ≤ (𝐸 · 𝑗))
161	54, 18, 26, 160	leadd1dd 11690	. . . 4 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝑗 − 1))(𝐸 · ((𝑋‘𝑖)‘𝑡)) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ≤ ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))))
162	14, 55, 27, 131, 161	ltletrd 11236	. . 3 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))))
163	8, 8	remulcld 11106	. . . . 5 ⊢ (𝜑 → (𝐸 · 𝐸) ∈ ℝ)
164	18, 163	readdcld 11105	. . . 4 ⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) ∈ ℝ)
165	60, 75	subcld 11433	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − 𝑗) ∈ ℂ)
166	165, 61	addcld 11095	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 𝑗) + 1) ∈ ℂ)
167	80, 166, 121	mul12d 11285	. . . . . 6 ⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) = (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁))))
168	167	oveq2d 7353	. . . . 5 ⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)))) = ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))))
169	23, 24	remulcld 11106	. . . . . . 7 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) ∈ ℝ)
170	8, 169	remulcld 11106	. . . . . 6 ⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) ∈ ℝ)
171	166, 80, 60, 120	div12d 11888	. . . . . . . 8 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) = (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)))
172	22, 17	resubcld 11504	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑗) ∈ ℝ)
173		elfzle1 13360	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1...𝑁) → 1 ≤ 𝑗)
174	15, 173	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑗)
175	22, 17	suble0d 11667	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 − 𝑗) ≤ 0 ↔ 1 ≤ 𝑗))
176	174, 175	mpbird 256	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 − 𝑗) ≤ 0)
177	172, 88, 20, 176	leadd2dd 11691	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 + (1 − 𝑗)) ≤ (𝑁 + 0))
178	60, 61, 75	addsub12d 11456	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 + (1 − 𝑗)) = (1 + (𝑁 − 𝑗)))
179	61, 165	addcomd 11278	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 + (𝑁 − 𝑗)) = ((𝑁 − 𝑗) + 1))
180	178, 179	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 + (1 − 𝑗)) = ((𝑁 − 𝑗) + 1))
181	60	addid1d 11276	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 + 0) = 𝑁)
182	177, 180, 181	3brtr3d 5123	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 − 𝑗) + 1) ≤ 𝑁)
183	19	nngt0d 12123	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 𝑁)
184		lediv1 11941	. . . . . . . . . . . . 13 ⊢ ((((𝑁 − 𝑗) + 1) ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ (𝑁 ∈ ℝ ∧ 0 < 𝑁)) → (((𝑁 − 𝑗) + 1) ≤ 𝑁 ↔ (((𝑁 − 𝑗) + 1) / 𝑁) ≤ (𝑁 / 𝑁)))
185	23, 20, 20, 183, 184	syl112anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) ≤ 𝑁 ↔ (((𝑁 − 𝑗) + 1) / 𝑁) ≤ (𝑁 / 𝑁)))
186	182, 185	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) / 𝑁) ≤ (𝑁 / 𝑁))
187	60, 120	dividd 11850	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 / 𝑁) = 1)
188	186, 187	breqtrd 5118	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) / 𝑁) ≤ 1)
189	23, 19	nndivred 12128	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) / 𝑁) ∈ ℝ)
190	189, 22, 7	lemul2d 12917	. . . . . . . . . 10 ⊢ (𝜑 → ((((𝑁 − 𝑗) + 1) / 𝑁) ≤ 1 ↔ (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)) ≤ (𝐸 · 1)))
191	188, 190	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)) ≤ (𝐸 · 1))
192	191, 139	breqtrd 5118	. . . . . . . 8 ⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) / 𝑁)) ≤ 𝐸)
193	171, 192	eqbrtrd 5114	. . . . . . 7 ⊢ (𝜑 → (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) ≤ 𝐸)
194	169, 8, 7	lemul2d 12917	. . . . . . 7 ⊢ (𝜑 → ((((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)) ≤ 𝐸 ↔ (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) ≤ (𝐸 · 𝐸)))
195	193, 194	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁))) ≤ (𝐸 · 𝐸))
196	170, 163, 18, 195	leadd2dd 11691	. . . . 5 ⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · (((𝑁 − 𝑗) + 1) · (𝐸 / 𝑁)))) ≤ ((𝐸 · 𝑗) + (𝐸 · 𝐸)))
197	168, 196	eqbrtrrd 5116	. . . 4 ⊢ (𝜑 → ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) ≤ ((𝐸 · 𝑗) + (𝐸 · 𝐸)))
198	80, 75	mulcomd 11097	. . . . . . 7 ⊢ (𝜑 → (𝐸 · 𝑗) = (𝑗 · 𝐸))
199	198	oveq1d 7352	. . . . . 6 ⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) = ((𝑗 · 𝐸) + (𝐸 · 𝐸)))
200	75, 80, 80	adddird 11101	. . . . . 6 ⊢ (𝜑 → ((𝑗 + 𝐸) · 𝐸) = ((𝑗 · 𝐸) + (𝐸 · 𝐸)))
201	199, 200	eqtr4d 2779	. . . . 5 ⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) = ((𝑗 + 𝐸) · 𝐸))
202	17, 8	readdcld 11105	. . . . . 6 ⊢ (𝜑 → (𝑗 + 𝐸) ∈ ℝ)
203		stoweidlem11.8	. . . . . . 7 ⊢ (𝜑 → 𝐸 < (1 / 3))
204	8, 32, 17, 203	ltadd2dd 11235	. . . . . 6 ⊢ (𝜑 → (𝑗 + 𝐸) < (𝑗 + (1 / 3)))
205	202, 33, 7, 204	ltmul1dd 12928	. . . . 5 ⊢ (𝜑 → ((𝑗 + 𝐸) · 𝐸) < ((𝑗 + (1 / 3)) · 𝐸))
206	201, 205	eqbrtrd 5114	. . . 4 ⊢ (𝜑 → ((𝐸 · 𝑗) + (𝐸 · 𝐸)) < ((𝑗 + (1 / 3)) · 𝐸))
207	27, 164, 34, 197, 206	lelttrd 11234	. . 3 ⊢ (𝜑 → ((𝐸 · 𝑗) + (((𝑁 − 𝑗) + 1) · (𝐸 · (𝐸 / 𝑁)))) < ((𝑗 + (1 / 3)) · 𝐸))
208	14, 27, 34, 162, 207	lttrd 11237	. 2 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)) < ((𝑗 + (1 / 3)) · 𝐸))
209	5, 208	eqbrtrd 5114	1 ⊢ (𝜑 → ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑁)(𝐸 · ((𝑋‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 Vcvv 3441 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4269 class class class wbr 5092 ↦ cmpt 5175 ⟶wf 6475 ‘cfv 6479 (class class class)co 7337 Fincfn 8804 ℂcc 10970 ℝcr 10971 0cc0 10972 1c1 10973 + caddc 10975 · cmul 10977 < clt 11110 ≤ cle 11111 − cmin 11306 / cdiv 11733 ℕcn 12074 3c3 12130 ℤcz 12420 ℤ_≥cuz 12683 ℝ⁺crp 12831 ...cfz 13340 ♯chash 14145 Σcsu 15496

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-inf2 9498 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-pre-sup 11050

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-se 5576 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-isom 6488 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-sup 9299 df-oi 9367 df-card 9796 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-div 11734 df-nn 12075 df-2 12137 df-3 12138 df-n0 12335 df-z 12421 df-uz 12684 df-rp 12832 df-ico 13186 df-fz 13341 df-fzo 13484 df-seq 13823 df-exp 13884 df-hash 14146 df-cj 14909 df-re 14910 df-im 14911 df-sqrt 15045 df-abs 15046 df-clim 15296 df-sum 15497

This theorem is referenced by: stoweidlem34 43919

W3C validator