stoweidlem42 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem42 43212

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 10817	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
3		stoweidlem42.11	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 12611	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	2, 4	resubcld 11243	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ∈ ℝ)
6	5	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ∈ ℝ)
7		stoweidlem42.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	4, 7	nndivred 11867	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℝ)
9	2, 8	resubcld 11243	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ)
10	9	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ)
11	7	nnnn0d 12133	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
12	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ₀)
13	10, 12	reexpcld 13716	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) ∈ ℝ)
14		elnnuz 12461	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
15	7, 14	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
16	15	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘1))
17		stoweidlem42.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
18		nfv 1922	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑡 ∈ 𝐵
19	17, 18	nfan 1907	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐵)
20		nfv 1922	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑀)
21	19, 20	nfan 1907	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))
22		stoweidlem42.6	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
23		nfcv 2900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑇
24		nfmpt1 5142	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
25	23, 24	nfmpt 5141	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
26	22, 25	nfcxfr 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝐹
27		nfcv 2900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑡
28	26, 27	nffv 6716	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑡)
29		nfcv 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑎
30	28, 29	nffv 6716	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎)
31	30	nfel1 2916	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎) ∈ ℝ
32	21, 31	nfim 1904	. . . . . . . 8 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
33		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑀) ↔ 𝑎 ∈ (1...𝑀)))
34	33	anbi2d 632	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))))
35		fveq2 6706	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑎))
36	35	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝐹‘𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹‘𝑡)‘𝑎) ∈ ℝ))
37	34, 36	imbi12d 348	. . . . . . . 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 3891	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
40		ovex 7235	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ V
41		mptexg 7026	. . . . . . . . . . . 12 ⊢ ((1...𝑀) ∈ V → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V)
42	40, 41	mp1i 13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V)
43	22	fvmpt2 6818	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
44	39, 42, 43	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
45		stoweidlem42.9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
46	45	ffvelrnda 6893	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
47		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
48	47, 46	jca 515	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌))
49		eleq1 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝑌 ↔ (𝑈‘𝑖) ∈ 𝑌))
50	49	anbi2d 632	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌)))
51		feq1 6515	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
52	50, 51	imbi12d 348	. . . . . . . . . . . . . 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 3474	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑖) ∈ 𝑌 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌) → (𝑈‘𝑖):𝑇⟶ℝ))
55	46, 48, 54	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
56	55	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
57	39	adantr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
58	56, 57	ffvelrnd 6894	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
59	44, 58	fvmpt2d 6820	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡))
60	59, 58	eqeltrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ)
61	32, 37, 60	chvarfv 2238	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
62		remulcl 10797	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
63	62	adantl 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
64	16, 61, 63	seqcl 13579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
65	3	rpcnd 12613	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
66	7	nncnd 11829	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67	7	nnne0d 11863	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ≠ 0)
68	65, 66, 67	divcan1d 11592	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) = 𝐸)
69	68	eqcomd 2740	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 = ((𝐸 / 𝑀) · 𝑀))
70	69	oveq2d 7218	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝐸) = (1 − ((𝐸 / 𝑀) · 𝑀)))
71		1cnd 10811	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
72	65, 66, 67	divcld 11591	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℂ)
73	72, 66	mulcld 10836	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) ∈ ℂ)
74	71, 73	negsubd 11178	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 − ((𝐸 / 𝑀) · 𝑀)))
75	72, 66	mulneg1d 11268	. . . . . . . . . . 11 ⊢ (𝜑 → (-(𝐸 / 𝑀) · 𝑀) = -((𝐸 / 𝑀) · 𝑀))
76	75	eqcomd 2740	. . . . . . . . . 10 ⊢ (𝜑 → -((𝐸 / 𝑀) · 𝑀) = (-(𝐸 / 𝑀) · 𝑀))
77	76	oveq2d 7218	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 + (-(𝐸 / 𝑀) · 𝑀)))
78	70, 74, 77	3eqtr2d 2780	. . . . . . . 8 ⊢ (𝜑 → (1 − 𝐸) = (1 + (-(𝐸 / 𝑀) · 𝑀)))
79	8	renegcld 11242	. . . . . . . . . 10 ⊢ (𝜑 → -(𝐸 / 𝑀) ∈ ℝ)
80	7	nnred 11828	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
81		3re 11893	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ∈ ℝ)
83		3ne0 11919	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ≠ 0)
85	82, 84	rereccld 11642	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) ∈ ℝ)
86		stoweidlem42.12	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 < (1 / 3))
87		1lt3 11986	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 3
88	87	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 3)
89		0lt1 11337	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 1
90	89	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 1)
91		3pos 11918	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 3
92	91	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 3)
93		ltdiv2 11701	. . . . . . . . . . . . . . . . . . 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 1388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 < 3 ↔ (1 / 3) < (1 / 1)))
95	88, 94	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 / 3) < (1 / 1))
96		1div1e1 11505	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
97	95, 96	breqtrdi 5084	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) < 1)
98	4, 85, 2, 86, 97	lttrd 10976	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 < 1)
99	7	nnge1d 11861	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑀)
100	4, 2, 80, 98, 99	ltletrd 10975	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 < 𝑀)
101	4, 80, 100	ltled 10963	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≤ 𝑀)
102	3	rpregt0d 12617	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
103	7	nngt0d 11862	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑀)
104		lediv2 11705	. . . . . . . . . . . . . 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 1377	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 𝐸)))
106	101, 105	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 𝐸))
107	3	rpcnne0d 12620	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
108		divid 11502	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ 𝐸 ≠ 0) → (𝐸 / 𝐸) = 1)
109	107, 108	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝐸) = 1)
110	106, 109	breqtrd 5069	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 1)
111	8, 2	lenegd 11394	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) ≤ 1 ↔ -1 ≤ -(𝐸 / 𝑀)))
112	110, 111	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → -1 ≤ -(𝐸 / 𝑀))
113		bernneq 13779	. . . . . . . . . 10 ⊢ ((-(𝐸 / 𝑀) ∈ ℝ ∧ 𝑀 ∈ ℕ₀ ∧ -1 ≤ -(𝐸 / 𝑀)) → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀))
114	79, 11, 112, 113	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀))
115	71, 72	negsubd 11178	. . . . . . . . . 10 ⊢ (𝜑 → (1 + -(𝐸 / 𝑀)) = (1 − (𝐸 / 𝑀)))
116	115	oveq1d 7217	. . . . . . . . 9 ⊢ (𝜑 → ((1 + -(𝐸 / 𝑀))↑𝑀) = ((1 − (𝐸 / 𝑀))↑𝑀))
117	114, 116	breqtrd 5069	. . . . . . . 8 ⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
118	78, 117	eqbrtrd 5065	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
119	118	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
120		eqid 2734	. . . . . . 7 ⊢ seq1( · , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡))
121	7	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ)
122		eqid 2734	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
123	19, 58, 122	fmptdf 6923	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
124	44	feq1d 6519	. . . . . . . 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 3123	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
128	127	an32s 652	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
129	128, 59	breqtrrd 5071	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝐹‘𝑡)‘𝑖))
130	72	addid2d 11016	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) = (𝐸 / 𝑀))
131		lediv2 11705	. . . . . . . . . . . . . . 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 1383	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
133	99, 132	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
134	65	div1d 11583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
135	133, 134	breqtrd 5069	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
136	8, 4, 2, 135, 98	lelttrd 10973	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) < 1)
137	130, 136	eqbrtrd 5065	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) < 1)
138		0red 10819	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
139	138, 8, 2	ltaddsubd 11415	. . . . . . . . . 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 12608	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
142	141	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
143	28, 19, 120, 121, 125, 129, 142	stoweidlem3 43173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) < (seq1( · , (𝐹‘𝑡))‘𝑀))
144	6, 13, 64, 119, 143	lelttrd 10973	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (seq1( · , (𝐹‘𝑡))‘𝑀))
145		stoweidlem42.7	. . . . . . 7 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
146	145	fvmpt2 6818	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
147	39, 64, 146	syl2anc 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
148	144, 147	breqtrrd 5071	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑍‘𝑡))
149		simpl 486	. . . . 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 42753	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
156	149, 39, 155	syl2anc 587	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑋‘𝑡) = (𝑍‘𝑡))
157	148, 156	breqtrrd 5071	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑋‘𝑡))
158	157	ex 416	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝑋‘𝑡)))
159	1, 158	ralrimi 3130	1 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2880 ≠ wne 2935 ∀wral 3054 Vcvv 3401 ⊆ wss 3857 class class class wbr 5043 ↦ cmpt 5124 ⟶wf 6365 ‘cfv 6369 (class class class)co 7202 ∈ cmpo 7204 ℂcc 10710 ℝcr 10711 0cc0 10712 1c1 10713 + caddc 10715 · cmul 10717 < clt 10850 ≤ cle 10851 − cmin 11045 -cneg 11046 / cdiv 11472 ℕcn 11813 3c3 11869 ℕ₀cn0 12073 ℤ_≥cuz 12421 ℝ⁺crp 12569 ...cfz 13078 seqcseq 13557 ↑cexp 13618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-n0 12074 df-z 12160 df-uz 12422 df-rp 12570 df-fz 13079 df-fzo 13222 df-seq 13558 df-exp 13619

This theorem is referenced by: stoweidlem51 43221

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