stoweidlem42 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem42		Structured version Visualization version GIF version

Theorem stoweidlem42 47325

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 11348	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
3		stoweidlem42.11	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 13204	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	2, 4	resubcld 11783	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ∈ ℝ)
6	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ∈ ℝ)
7		stoweidlem42.8	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	4, 7	nndivred 12432	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℝ)
9	2, 8	resubcld 11783	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ)
10	9	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ)
11	7	nnnn0d 12707	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
12	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ₀)
13	10, 12	reexpcld 14346	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) ∈ ℝ)
14		elnnuz 13045	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (ℤ_≥‘1))
15	7, 14	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
16	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ (ℤ_≥‘1))
17		stoweidlem42.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝜑
18		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑖 𝑡 ∈ 𝐵
19	17, 18	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑡 ∈ 𝐵)
20		nfv 1949	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑎 ∈ (1...𝑀)
21	19, 20	nfan 1934	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))
22		stoweidlem42.6	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
23		nfcv 2926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖𝑇
24		nfmpt1 5210	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
25	23, 24	nfmpt 5209	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑖(𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
26	22, 25	nfcxfr 2924	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝐹
27		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖𝑡
28	26, 27	nffv 6926	. . . . . . . . . . 11 ⊢ Ⅎ𝑖(𝐹‘𝑡)
29		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑖𝑎
30	28, 29	nffv 6926	. . . . . . . . . 10 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎)
31	30	nfel1 2942	. . . . . . . . 9 ⊢ Ⅎ𝑖((𝐹‘𝑡)‘𝑎) ∈ ℝ
32	21, 31	nfim 1931	. . . . . . . 8 ⊢ Ⅎ𝑖(((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
33		eleq1 2852	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → (𝑖 ∈ (1...𝑀) ↔ 𝑎 ∈ (1...𝑀)))
34	33	anbi2d 642	. . . . . . . . 9 ⊢ (𝑖 = 𝑎 → (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) ↔ ((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀))))
35		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑖 = 𝑎 → ((𝐹‘𝑡)‘𝑖) = ((𝐹‘𝑡)‘𝑎))
36	35	eleq1d 2849	. . . . . . . . 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 3913	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑡 ∈ 𝑇)
40		ovex 7484	. . . . . . . . . . . 12 ⊢ (1...𝑀) ∈ V
41		mptexg 7258	. . . . . . . . . . . 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 7036	. . . . . . . . . . 11 ⊢ ((𝑡 ∈ 𝑇 ∧ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) ∈ V) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
44	39, 42, 43	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝐹‘𝑡) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
45		stoweidlem42.9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
46	45	ffvelcdmda 7116	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
47		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
48	47, 46	jca 521	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌))
49		eleq1 2852	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝑌 ↔ (𝑈‘𝑖) ∈ 𝑌))
50	49	anbi2d 642	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝑌) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝑌)))
51		feq1 6717	. . . . . . . . . . . . . . 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 3521	. . . . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
57	39	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → 𝑡 ∈ 𝑇)
58	56, 57	ffvelcdmd 7117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
59	44, 58	fvmpt2d 7038	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) = ((𝑈‘𝑖)‘𝑡))
60	59, 58	eqeltrd 2864	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑖) ∈ ℝ)
61	32, 37, 60	chvarfv 2280	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑎 ∈ (1...𝑀)) → ((𝐹‘𝑡)‘𝑎) ∈ ℝ)
62		remulcl 11324	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑎 · 𝑗) ∈ ℝ)
63	62	adantl 487	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ (𝑎 ∈ ℝ ∧ 𝑗 ∈ ℝ)) → (𝑎 · 𝑗) ∈ ℝ)
64	16, 61, 63	seqcl 14205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ)
65	3	rpcnd 13206	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ ℂ)
66	7	nncnd 12391	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ ℂ)
67	7	nnne0d 12428	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ≠ 0)
68	65, 66, 67	divcan1d 12134	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) = 𝐸)
69	68	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 = ((𝐸 / 𝑀) · 𝑀))
70	69	oveq2d 7467	. . . . . . . . 9 ⊢ (𝜑 → (1 − 𝐸) = (1 − ((𝐸 / 𝑀) · 𝑀)))
71		1cnd 11341	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℂ)
72	65, 66, 67	divcld 12133	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ∈ ℂ)
73	72, 66	mulcld 11368	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 / 𝑀) · 𝑀) ∈ ℂ)
74	71, 73	negsubd 11714	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 − ((𝐸 / 𝑀) · 𝑀)))
75	72, 66	mulneg1d 11808	. . . . . . . . . . 11 ⊢ (𝜑 → ( -(𝐸 / 𝑀) · 𝑀) = -((𝐸 / 𝑀) · 𝑀))
76	75	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝜑 → -((𝐸 / 𝑀) · 𝑀) = ( -(𝐸 / 𝑀) · 𝑀))
77	76	oveq2d 7467	. . . . . . . . 9 ⊢ (𝜑 → (1 + -((𝐸 / 𝑀) · 𝑀)) = (1 + ( -(𝐸 / 𝑀) · 𝑀)))
78	70, 74, 77	3eqtr2d 2804	. . . . . . . 8 ⊢ (𝜑 → (1 − 𝐸) = (1 + ( -(𝐸 / 𝑀) · 𝑀)))
79	8	renegcld 11782	. . . . . . . . . 10 ⊢ (𝜑 → -(𝐸 / 𝑀) ∈ ℝ)
80	7	nnred 12390	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℝ)
81		3re 12463	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ∈ ℝ
82	81	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ∈ ℝ)
83		3ne0 12492	. . . . . . . . . . . . . . . . . 18 ⊢ 3 ≠ 0
84	83	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 3 ≠ 0)
85	82, 84	rereccld 12184	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) ∈ ℝ)
86		stoweidlem42.12	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 < (1 / 3))
87		1lt3 12558	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 < 3
88	87	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 1 < 3)
89		0lt1 11878	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 1
90	89	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 1)
91		3pos 12491	. . . . . . . . . . . . . . . . . . . 20 ⊢ 0 < 3
92	91	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 0 < 3)
93		ltdiv2 12243	. . . . . . . . . . . . . . . . . . 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 1414	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1 < 3 ↔ (1 / 3) < (1 / 1)))
95	88, 94	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1 / 3) < (1 / 1))
96		1div1e1 12047	. . . . . . . . . . . . . . . . 17 ⊢ (1 / 1) = 1
97	95, 96	breqtrdi 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 / 3) < 1)
98	4, 85, 2, 86, 97	lttrd 11510	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 < 1)
99	7	nnge1d 12426	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ 𝑀)
100	4, 2, 80, 98, 99	ltletrd 11509	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 < 𝑀)
101	4, 80, 100	ltled 11497	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≤ 𝑀)
102	3	rpregt0d 13210	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
103	7	nngt0d 12427	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < 𝑀)
104		lediv2 12247	. . . . . . . . . . . . . 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 1403	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 𝐸)))
106	101, 105	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 𝐸))
107	3	rpcnne0d 13213	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 ∈ ℂ ∧ 𝐸 ≠ 0))
108		divid 12044	. . . . . . . . . . . . 13 ⊢ ((𝐸 ∈ ℂ ∧ 𝐸 ≠ 0) → (𝐸 / 𝐸) = 1)
109	107, 108	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝐸) = 1)
110	106, 109	breqtrd 5137	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 1)
111	8, 2	lenegd 11935	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 / 𝑀) ≤ 1 ↔ -1 ≤ -(𝐸 / 𝑀)))
112	110, 111	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → -1 ≤ -(𝐸 / 𝑀))
113		bernneq 14414	. . . . . . . . . 10 ⊢ (( -(𝐸 / 𝑀) ∈ ℝ ∧ 𝑀 ∈ ℕ₀ ∧ -1 ≤ -(𝐸 / 𝑀)) → (1 + ( -(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀))
114	79, 11, 112, 113	syl3anc 1399	. . . . . . . . 9 ⊢ (𝜑 → (1 + ( -(𝐸 / 𝑀) · 𝑀)) ≤ ((1 + -(𝐸 / 𝑀))↑𝑀))
115	71, 72	negsubd 11714	. . . . . . . . . 10 ⊢ (𝜑 → (1 + -(𝐸 / 𝑀)) = (1 − (𝐸 / 𝑀)))
116	115	oveq1d 7466	. . . . . . . . 9 ⊢ (𝜑 → ((1 + -(𝐸 / 𝑀))↑𝑀) = ((1 − (𝐸 / 𝑀))↑𝑀))
117	114, 116	breqtrd 5137	. . . . . . . 8 ⊢ (𝜑 → (1 + ( -(𝐸 / 𝑀) · 𝑀)) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
118	78, 117	eqbrtrd 5133	. . . . . . 7 ⊢ (𝜑 → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
119	118	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) ≤ ((1 − (𝐸 / 𝑀))↑𝑀))
120		eqid 2763	. . . . . . 7 ⊢ seq1( · , (𝐹‘𝑡)) = seq1( · , (𝐹‘𝑡))
121	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → 𝑀 ∈ ℕ)
122		eqid 2763	. . . . . . . . 9 ⊢ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)) = (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡))
123	19, 58, 122	fmptdf 7149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)):(1...𝑀)⟶ℝ)
124	44	feq1d 6721	. . . . . . . 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 3258	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
128	127	an32s 665	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
129	128, 59	breqtrrd 5139	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑀)) → (1 − (𝐸 / 𝑀)) < ((𝐹‘𝑡)‘𝑖))
130	72	addlidd 11550	. . . . . . . . . . 11 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) = (𝐸 / 𝑀))
131		lediv2 12247	. . . . . . . . . . . . . . 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 1409	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
133	99, 132	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
134	65	div1d 12125	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
135	133, 134	breqtrd 5137	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
136	8, 4, 2, 135, 98	lelttrd 11507	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 / 𝑀) < 1)
137	130, 136	eqbrtrd 5133	. . . . . . . . . 10 ⊢ (𝜑 → (0 + (𝐸 / 𝑀)) < 1)
138		0red 11350	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℝ)
139	138, 8, 2	ltaddsubd 11956	. . . . . . . . . 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 13201	. . . . . . . 8 ⊢ (𝜑 → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
142	141	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − (𝐸 / 𝑀)) ∈ ℝ⁺)
143	28, 19, 120, 121, 125, 129, 142	stoweidlem3 47286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → ((1 − (𝐸 / 𝑀))↑𝑀) < (seq1( · , (𝐹‘𝑡))‘𝑀))
144	6, 13, 64, 119, 143	lelttrd 11507	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (seq1( · , (𝐹‘𝑡))‘𝑀))
145		stoweidlem42.7	. . . . . . 7 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
146	145	fvmpt2 7036	. . . . . 6 ⊢ ((𝑡 ∈ 𝑇 ∧ (seq1( · , (𝐹‘𝑡))‘𝑀) ∈ ℝ) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
147	39, 64, 146	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑍‘𝑡) = (seq1( · , (𝐹‘𝑡))‘𝑀))
148	144, 147	breqtrrd 5139	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑍‘𝑡))
149		simpl 488	. . . . 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 46868	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (𝑍‘𝑡))
156	149, 39, 155	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (𝑋‘𝑡) = (𝑍‘𝑡))
157	148, 156	breqtrrd 5139	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐵) → (1 − 𝐸) < (𝑋‘𝑡))
158	157	ex 418	. 2 ⊢ (𝜑 → (𝑡 ∈ 𝐵 → (1 − 𝐸) < (𝑋‘𝑡)))
159	1, 158	ralrimi 3264	1 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2147 Ⅎwnfc 2911 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 class class class wbr 5108 ↦ cmpt 5192 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 < clt 11382 ≤ cle 11383 − cmin 11580 -cneg 11581 / cdiv 12013 ℕcn 12375 3c3 12438 ℕ₀cn0 12646 ℤ_≥cuz 13005 ℝ⁺crp 13160 ...cfz 13679 seqcseq 14184 ↑cexp 14244

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-z 12734 df-uz 13006 df-rp 13161 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245

This theorem is used by: stoweidlem51 47334

W3C validator