stoweidlem42 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem42 46033

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 11116	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
3		stoweidlem42.11	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 12937	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	2, 4	resubcld 11548	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ∈ ℝ)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ∈ ℝ)
7		stoweidlem42.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	4, 7	nndivred 12182	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℝ)
9	2, 8	resubcld 11548	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ)
10	9	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ)
11	7	nnnn0d 12445	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ₀)
13	10, 12	reexpcld 14070	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) ∈ ℝ)
14		elnnuz 12779	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
15	7, 14	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
16	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘1))
17		stoweidlem42.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
18		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑡 ∈ 𝐵
19	17, 18	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐵)
20		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑀)
21	19, 20	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))
22		stoweidlem42.6	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
23		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑇
24		nfmpt1 5191	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
25	23, 24	nfmpt 5190	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
26	22, 25	nfcxfr 2889	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝐹
27		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑡
28	26, 27	nffv 6832	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑡)
29		nfcv 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑎
30	28, 29	nffv 6832	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎)
31	30	nfel1 2908	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎) ∈ ℝ
32	21, 31	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
33		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑀) ↔ 𝑎 ∈ (1...𝑀)))
34	33	anbi2d 630	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))))
35		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑎))
36	35	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝐹‘𝑡)‘𝑖) ∈ ℝ ↔ ((𝐹‘𝑡)‘𝑎) ∈ ℝ))
37	34, 36	imbi12d 344	. . . . . . . 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 3935	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
40		ovex 7382	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ V
41		mptexg 7157	. . . . . . . . . . . 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 6941	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
44	39, 42, 43	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
45		stoweidlem42.9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
46	45	ffvelcdmda 7018	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
47		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
48	47, 46	jca 511	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌))
49		eleq1 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝑌 ↔ (𝑈‘𝑖) ∈ 𝑌))
50	49	anbi2d 630	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌)))
51		feq1 6630	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
52	50, 51	imbi12d 344	. . . . . . . . . . . . . 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 3509	. . . . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
58	56, 57	ffvelcdmd 7019	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
59	44, 58	fvmpt2d 6943	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡))
60	59, 58	eqeltrd 2828	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ)
61	32, 37, 60	chvarfv 2241	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
62		remulcl 11094	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
63	62	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
64	16, 61, 63	seqcl 13929	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
65	3	rpcnd 12939	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
66	7	nncnd 12144	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67	7	nnne0d 12178	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ≠ 0)
68	65, 66, 67	divcan1d 11901	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) = 𝐸)
69	68	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 = ((𝐸 / 𝑀) · 𝑀))
70	69	oveq2d 7365	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝐸) = (1 − ((𝐸 / 𝑀) · 𝑀)))
71		1cnd 11110	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
72	65, 66, 67	divcld 11900	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℂ)
73	72, 66	mulcld 11135	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) ∈ ℂ)
74	71, 73	negsubd 11481	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 − ((𝐸 / 𝑀) · 𝑀)))
75	72, 66	mulneg1d 11573	. . . . . . . . . . 11 ⊢ (𝜑 → (-(𝐸 / 𝑀) · 𝑀) = -((𝐸 / 𝑀) · 𝑀))
76	75	eqcomd 2735	. . . . . . . . . 10 ⊢ (𝜑 → -((𝐸 / 𝑀) · 𝑀) = (-(𝐸 / 𝑀) · 𝑀))
77	76	oveq2d 7365	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 + (-(𝐸 / 𝑀) · 𝑀)))
78	70, 74, 77	3eqtr2d 2770	. . . . . . . 8 ⊢ (𝜑 → (1 − 𝐸) = (1 + (-(𝐸 / 𝑀) · 𝑀)))
79	8	renegcld 11547	. . . . . . . . . 10 ⊢ (𝜑 → -(𝐸 / 𝑀) ∈ ℝ)
80	7	nnred 12143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
81		3re 12208	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ∈ ℝ)
83		3ne0 12234	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ≠ 0)
85	82, 84	rereccld 11951	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) ∈ ℝ)
86		stoweidlem42.12	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 < (1 / 3))
87		1lt3 12296	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 3
88	87	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 3)
89		0lt1 11642	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 1
90	89	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 1)
91		3pos 12233	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 3
92	91	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 3)
93		ltdiv2 12011	. . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 / 3) < (1 / 1))
96		1div1e1 11815	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
97	95, 96	breqtrdi 5133	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) < 1)
98	4, 85, 2, 86, 97	lttrd 11277	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 < 1)
99	7	nnge1d 12176	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑀)
100	4, 2, 80, 98, 99	ltletrd 11276	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 < 𝑀)
101	4, 80, 100	ltled 11264	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≤ 𝑀)
102	3	rpregt0d 12943	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
103	7	nngt0d 12177	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑀)
104		lediv2 12015	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 𝐸))
107	3	rpcnne0d 12946	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
108		divid 11810	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ 𝐸 ≠ 0) → (𝐸 / 𝐸) = 1)
109	107, 108	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝐸) = 1)
110	106, 109	breqtrd 5118	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 1)
111	8, 2	lenegd 11699	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) ≤ 1 ↔ -1 ≤ -(𝐸 / 𝑀)))
112	110, 111	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → -1 ≤ -(𝐸 / 𝑀))
113		bernneq 14136	. . . . . . . . . 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 11481	. . . . . . . . . 10 ⊢ (𝜑 → (1 + -(𝐸 / 𝑀)) = (1 − (𝐸 / 𝑀)))
116	115	oveq1d 7364	. . . . . . . . 9 ⊢ (𝜑 → ((1 + -(𝐸 / 𝑀))↑𝑀) = ((1 − (𝐸 / 𝑀))↑𝑀))
117	114, 116	breqtrd 5118	. . . . . . . 8 ⊢ (𝜑 → (1 + (-(𝐸 / 𝑀) · 𝑀)) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
118	78, 117	eqbrtrd 5114	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
119	118	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
120		eqid 2729	. . . . . . 7 ⊢ seq1( · , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡))
121	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ)
122		eqid 2729	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
123	19, 58, 122	fmptdf 7051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
124	44	feq1d 6634	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((𝐹‘𝑡):(1...𝑀)⟶ℝ ↔ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ))
125	123, 124	mpbird 257	. . . . . . 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 3221	. . . . . . . . 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 5120	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝐹‘𝑡)‘𝑖))
130	72	addlidd 11317	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) = (𝐸 / 𝑀))
131		lediv2 12015	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
134	65	div1d 11892	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
135	133, 134	breqtrd 5118	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
136	8, 4, 2, 135, 98	lelttrd 11274	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) < 1)
137	130, 136	eqbrtrd 5114	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) < 1)
138		0red 11118	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
139	138, 8, 2	ltaddsubd 11720	. . . . . . . . . 10 ⊢ (𝜑 → ((0 + (𝐸 / 𝑀)) < 1 ↔ 0 < (1 − (𝐸 / 𝑀))))
140	137, 139	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → 0 < (1 − (𝐸 / 𝑀)))
141	9, 140	elrpd 12934	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
142	141	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
143	28, 19, 120, 121, 125, 129, 142	stoweidlem3 45994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) < (seq1( · , (𝐹‘𝑡))‘𝑀))
144	6, 13, 64, 119, 143	lelttrd 11274	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (seq1( · , (𝐹‘𝑡))‘𝑀))
145		stoweidlem42.7	. . . . . . 7 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
146	145	fvmpt2 6941	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
147	39, 64, 146	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
148	144, 147	breqtrrd 5120	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑍‘𝑡))
149		simpl 482	. . . . 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 45574	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
156	149, 39, 155	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑋‘𝑡) = (𝑍‘𝑡))
157	148, 156	breqtrrd 5120	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑋‘𝑡))
158	157	ex 412	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝑋‘𝑡)))
159	1, 158	ralrimi 3227	1 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2876 ≠ wne 2925 ∀wral 3044 Vcvv 3436 ⊆ wss 3903 class class class wbr 5092 ↦ cmpt 5173 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 < clt 11149 ≤ cle 11150 − cmin 11347 -cneg 11348 / cdiv 11777 ℕcn 12128 3c3 12184 ℕ₀cn0 12384 ℤ_≥cuz 12735 ℝ⁺crp 12893 ...cfz 13410 seqcseq 13908 ↑cexp 13968

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969

This theorem is referenced by: stoweidlem51 46042

W3C validator