stoweidlem42 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem42 46682

Description: This lemma is used to prove that 𝑥 built as in Lemma 2 of [BrosowskiDeutsh] p. 91, is such that x > 1 - ε on B. Here 𝑋 is used to represent 𝑥 in the paper, and E is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem42.1	⊢ Ⅎ𝑖𝜑
stoweidlem42.2	⊢ Ⅎ𝑡𝜑
stoweidlem42.3	⊢ Ⅎ𝑡𝑌
stoweidlem42.4	⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
stoweidlem42.5	⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
stoweidlem42.6	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
stoweidlem42.7	⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
stoweidlem42.8	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem42.9	⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
stoweidlem42.10	⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
stoweidlem42.11	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem42.12	⊢ (𝜑 → 𝐸 < (1 / 3))
stoweidlem42.13	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
stoweidlem42.14	⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
stoweidlem42.15	⊢ (𝜑 → 𝑇 ∈ V)
stoweidlem42.16	⊢ (𝜑 → 𝐵 ⊆ 𝑇)

**Assertion**
Ref	Expression
stoweidlem42	⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))

Distinct variable groups: 𝑡,𝑖 𝐵,𝑖 𝑖,𝑀 𝑓,𝑔,𝑡,𝑇 𝑓,𝑖,𝑇 𝑓,𝐹,𝑔 𝑓,𝑀,𝑔 𝑈,𝑓,𝑔,𝑡 𝑓,𝑌,𝑔 𝜑,𝑓,𝑔 𝑖,𝐸 𝑈,𝑖 Allowed substitution hints: 𝜑(𝑡,𝑖) 𝐵(𝑡,𝑓,𝑔) 𝑃(𝑡,𝑓,𝑔,𝑖) 𝐸(𝑡,𝑓,𝑔) 𝐹(𝑡,𝑖) 𝑀(𝑡) 𝑋(𝑡,𝑓,𝑔,𝑖) 𝑌(𝑡,𝑖) 𝑍(𝑡,𝑓,𝑔,𝑖)

Proof of Theorem stoweidlem42 Dummy variables 𝑎 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem42.2	. 2 ⊢ Ⅎ𝑡𝜑
2		1red 11209	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
3		stoweidlem42.11	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 13060	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	2, 4	resubcld 11642	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ∈ ℝ)
6	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ∈ ℝ)
7		stoweidlem42.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	4, 7	nndivred 12290	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℝ)
9	2, 8	resubcld 11642	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ)
10	9	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ)
11	7	nnnn0d 12565	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
12	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ₀)
13	10, 12	reexpcld 14199	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) ∈ ℝ)
14		elnnuz 12902	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
15	7, 14	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
16	15	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘1))
17		stoweidlem42.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
18		nfv 1941	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑡 ∈ 𝐵
19	17, 18	nfan 1926	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐵)
20		nfv 1941	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑀)
21	19, 20	nfan 1926	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))
22		stoweidlem42.6	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
23		nfcv 2931	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑇
24		nfmpt1 5214	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
25	23, 24	nfmpt 5213	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
26	22, 25	nfcxfr 2929	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝐹
27		nfcv 2931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑡
28	26, 27	nffv 6892	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑡)
29		nfcv 2931	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑎
30	28, 29	nffv 6892	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎)
31	30	nfel1 2947	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎) ∈ ℝ
32	21, 31	nfim 1923	. . . . . . . 8 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
33		eleq1 2857	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑀) ↔ 𝑎 ∈ (1...𝑀)))
34	33	anbi2d 641	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))))
35		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑎))
36	35	eleq1d 2854	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝐹‘𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹‘𝑡)‘𝑎) ∈ ℝ))
37	34, 36	imbi12d 347	. . . . . . . 8 ⊢ (𝑖 = 𝑎 → ((((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ) ↔ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)))
38		stoweidlem42.16	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
39	38	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
40		ovex 7444	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ V
41		mptexg 7220	. . . . . . . . . . . 12 ⊢ ((1...𝑀) ∈ V → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V)
42	40, 41	mp1i 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V)
43	22	fvmpt2 7002	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
44	39, 42, 43	syl2anc 595	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
45		stoweidlem42.9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
46	45	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
47		simpl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
48	47, 46	jca 520	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌))
49		eleq1 2857	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝑌 ↔ (𝑈‘𝑖) ∈ 𝑌))
50	49	anbi2d 641	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌)))
51		feq1 6684	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
52	50, 51	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ)))
53		stoweidlem42.13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
54	52, 53	vtoclg 3531	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ))
55	46, 48, 54	sylc 66	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
56	55	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
57	39	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
58	56, 57	ffvelcdmd 7081	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
59	44, 58	fvmpt2d 7004	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡))
60	59, 58	eqeltrd 2869	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ)
61	32, 37, 60	chvarfv 2282	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
62		remulcl 11185	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
63	62	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
64	16, 61, 63	seqcl 14058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
65	3	rpcnd 13062	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
66	7	nncnd 12249	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67	7	nnne0d 12286	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ≠ 0)
68	65, 66, 67	divcan1d 11992	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) = 𝐸)
69	68	eqcomd 2775	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 = ((𝐸 / 𝑀) · 𝑀))
70	69	oveq2d 7427	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝐸) = (1 − ((𝐸 / 𝑀) · 𝑀)))
71		1cnd 11202	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
72	65, 66, 67	divcld 11991	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℂ)
73	72, 66	mulcld 11229	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) ∈ ℂ)
74	71, 73	negsubd 11575	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 − ((𝐸 / 𝑀) · 𝑀)))
75	72, 66	mulneg1d 11667	. . . . . . . . . . 11 ⊢ (𝜑 → (-(𝐸 / 𝑀) · 𝑀) = -((𝐸 / 𝑀) · 𝑀))
76	75	eqcomd 2775	. . . . . . . . . 10 ⊢ (𝜑 → -((𝐸 / 𝑀) · 𝑀) = (-(𝐸 / 𝑀) · 𝑀))
77	76	oveq2d 7427	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 + (-(𝐸 / 𝑀) · 𝑀)))
78	70, 74, 77	3eqtr2d 2810	. . . . . . . 8 ⊢ (𝜑 → (1 − 𝐸) = (1 + (-(𝐸 / 𝑀) · 𝑀)))
79	8	renegcld 11641	. . . . . . . . . 10 ⊢ (𝜑 → -(𝐸 / 𝑀) ∈ ℝ)
80	7	nnred 12248	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
81		3re 12321	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ∈ ℝ)
83		3ne0 12350	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ≠ 0)
85	82, 84	rereccld 12042	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) ∈ ℝ)
86		stoweidlem42.12	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 < (1 / 3))
87		1lt3 12416	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 3
88	87	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 3)
89		0lt1 11736	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 1
90	89	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 1)
91		3pos 12349	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 3
92	91	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 3)
93		ltdiv2 12101	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (3 ∈ ℝ ∧ 0 < 3) ∧ (1 ∈ ℝ ∧ 0 < 1)) → (1 < 3 ↔ (1 / 3) < (1 / 1)))
94	2, 90, 82, 92, 2, 90, 93	syl222anc 1411	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 < 3 ↔ (1 / 3) < (1 / 1)))
95	88, 94	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 / 3) < (1 / 1))
96		1div1e1 11905	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
97	95, 96	breqtrdi 5156	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) < 1)
98	4, 85, 2, 86, 97	lttrd 11371	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 < 1)
99	7	nnge1d 12284	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑀)
100	4, 2, 80, 98, 99	ltletrd 11370	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 < 𝑀)
101	4, 80, 100	ltled 11358	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≤ 𝑀)
102	3	rpregt0d 13066	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
103	7	nngt0d 12285	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑀)
104		lediv2 12105	. . . . . . . . . . . . . 14 ⊢ (((𝐸 ∈ ℝ ∧ 0 < 𝐸) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (𝐸 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 𝐸)))
105	102, 80, 103, 102, 104	syl121anc 1400	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 𝐸)))
106	101, 105	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 𝐸))
107	3	rpcnne0d 13069	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
108		divid 11902	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ 𝐸 ≠ 0) → (𝐸 / 𝐸) = 1)
109	107, 108	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝐸) = 1)
110	106, 109	breqtrd 5141	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 1)
111	8, 2	lenegd 11793	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) ≤ 1 ↔ -1 ≤ -(𝐸 / 𝑀)))
112	110, 111	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → -1 ≤ -(𝐸 / 𝑀))
113		bernneq 14265	. . . . . . . . . 10 ⊢ ((-(𝐸 / 𝑀) ∈ ℝ ∧ 𝑀 ∈ ℕ₀ ∧ -1 ≤ -(𝐸 / 𝑀)) → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀))
114	79, 11, 112, 113	syl3anc 1396	. . . . . . . . 9 ⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀))
115	71, 72	negsubd 11575	. . . . . . . . . 10 ⊢ (𝜑 → (1 + -(𝐸 / 𝑀)) = (1 − (𝐸 / 𝑀)))
116	115	oveq1d 7426	. . . . . . . . 9 ⊢ (𝜑 → ((1 + -(𝐸 / 𝑀))↑𝑀) = ((1 − (𝐸 / 𝑀))↑𝑀))
117	114, 116	breqtrd 5141	. . . . . . . 8 ⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
118	78, 117	eqbrtrd 5137	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
119	118	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
120		eqid 2769	. . . . . . 7 ⊢ seq1( · , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡))
121	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ)
122		eqid 2769	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
123	19, 58, 122	fmptdf 7113	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
124	44	feq1d 6688	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((𝐹‘𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
125	123, 124	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡):(1...𝑀)⟶ℝ)
126		stoweidlem42.10	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
127	126	r19.21bi 3263	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
128	127	an32s 664	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
129	128, 59	breqtrrd 5143	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝐹‘𝑡)‘𝑖))
130	72	addlidd 11411	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) = (𝐸 / 𝑀))
131		lediv2 12105	. . . . . . . . . . . . . . 15 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
132	2, 90, 80, 103, 102, 131	syl221anc 1406	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
133	99, 132	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
134	65	div1d 11983	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
135	133, 134	breqtrd 5141	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
136	8, 4, 2, 135, 98	lelttrd 11368	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) < 1)
137	130, 136	eqbrtrd 5137	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) < 1)
138		0red 11211	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
139	138, 8, 2	ltaddsubd 11814	. . . . . . . . . 10 ⊢ (𝜑 → ((0 + (𝐸 / 𝑀)) < 1 ↔ 0 < (1 − (𝐸 / 𝑀))))
140	137, 139	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 − (𝐸 / 𝑀)))
141	9, 140	elrpd 13057	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
142	141	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
143	28, 19, 120, 121, 125, 129, 142	stoweidlem3 46643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) < (seq1( · , (𝐹‘𝑡))‘𝑀))
144	6, 13, 64, 119, 143	lelttrd 11368	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (seq1( · , (𝐹‘𝑡))‘𝑀))
145		stoweidlem42.7	. . . . . . 7 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
146	145	fvmpt2 7002	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
147	39, 64, 146	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
148	144, 147	breqtrrd 5143	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑍‘𝑡))
149		simpl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝜑)
150		stoweidlem42.3	. . . . . 6 ⊢ Ⅎ𝑡𝑌
151		stoweidlem42.4	. . . . . 6 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
152		stoweidlem42.5	. . . . . 6 ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
153		stoweidlem42.15	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V)
154		stoweidlem42.14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
155	17, 150, 151, 152, 22, 145, 153, 7, 45, 53, 154	fmuldfeq 46225	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
156	149, 39, 155	syl2anc 595	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑋‘𝑡) = (𝑍‘𝑡))
157	148, 156	breqtrrd 5143	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑋‘𝑡))
158	157	ex 417	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝑋‘𝑡)))
159	1, 158	ralrimi 3269	1 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 Ⅎwnf 1810 ∈ wcel 2149 Ⅎwnfc 2916 ≠ wne 2964 ∀wral 3085 Vcvv 3463 ⊆ wss 3913 class class class wbr 5113 ↦ cmpt 5196 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 ≤ cle 11244 − cmin 11441 -cneg 11442 / cdiv 11871 ℕcn 12233 3c3 12296 ℕ₀cn0 12504 ℤ_≥cuz 12862 ℝ⁺crp 13016 ...cfz 13535 seqcseq 14037 ↑cexp 14097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-er 8694 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-fz 13536 df-fzo 13683 df-seq 14038 df-exp 14098

This theorem is referenced by: stoweidlem51 46691

W3C validator