stoweidlem60 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem60 46700

Description: This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all 𝑡 in 𝑇, there is a 𝑗 such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem60.1	⊢ Ⅎ𝑡𝐹
stoweidlem60.2	⊢ Ⅎ𝑡𝜑
stoweidlem60.3	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem60.4	⊢ 𝑇 = ∪ 𝐽
stoweidlem60.5	⊢ 𝐶 = (𝐽 Cn 𝐾)
stoweidlem60.6	⊢ 𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
stoweidlem60.7	⊢ 𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)})
stoweidlem60.8	⊢ (𝜑 → 𝐽 ∈ Comp)
stoweidlem60.9	⊢ (𝜑 → 𝑇 ≠ ∅)
stoweidlem60.10	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
stoweidlem60.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem60.12	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem60.13	⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
stoweidlem60.14	⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
stoweidlem60.15	⊢ (𝜑 → 𝐹 ∈ 𝐶)
stoweidlem60.16	⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡))
stoweidlem60.17	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem60.18	⊢ (𝜑 → 𝐸 < (1 / 3))

**Assertion**
Ref	Expression
stoweidlem60	⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))

Distinct variable groups: 𝑓,𝑔,𝑗,𝑛,𝑡,𝐴,𝑞,𝑟 𝑦,𝑓,𝑗,𝑛,𝑞,𝑟,𝑡,𝐴 𝐵,𝑓,𝑔 𝐷,𝑓,𝑔 𝑓,𝐸,𝑔,𝑗,𝑛,𝑡 𝑓,𝐽,𝑔,𝑟,𝑡 𝑇,𝑓,𝑔,𝑗,𝑛,𝑡 𝜑,𝑓,𝑔,𝑗,𝑛 𝑔,𝐹,𝑗,𝑛 𝐵,𝑞,𝑟,𝑦 𝐷,𝑞,𝑟,𝑦 𝑇,𝑞,𝑟,𝑦 𝜑,𝑞,𝑟,𝑦 𝐸,𝑟,𝑦 𝑡,𝐾 Allowed substitution hints: 𝜑(𝑡) 𝐵(𝑡,𝑗,𝑛) 𝐶(𝑦,𝑡,𝑓,𝑔,𝑗,𝑛,𝑟,𝑞) 𝐷(𝑡,𝑗,𝑛) 𝐸(𝑞) 𝐹(𝑦,𝑡,𝑓,𝑟,𝑞) 𝐽(𝑦,𝑗,𝑛,𝑞) 𝐾(𝑦,𝑓,𝑔,𝑗,𝑛,𝑟,𝑞)

Proof of Theorem stoweidlem60 Dummy variables 𝑖 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnre 12240	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
2	1	adantl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3		stoweidlem60.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 13060	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	4	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
6	3	rpne0d 13065	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≠ 0)
7	6	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ≠ 0)
8	2, 5, 7	redivcld 12043	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9		1red 11209	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ)
10	8, 9	readdcld 11238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
11	10	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12		arch 12501	. . . . . . . . 9 ⊢ (((𝑚 / 𝐸) + 1) ∈ ℝ → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛)
13	11, 12	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛)
14		stoweidlem60.2	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝜑
15		nfv 1941	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
16	14, 15	nfan 1926	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
17		nfra1 3295	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚
18	16, 17	nfan 1926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
19		nfv 1941	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡 𝑛 ∈ ℕ
20	18, 19	nfan 1926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡(((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21		nfv 1941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑚 / 𝐸) + 1) < 𝑛
22	20, 21	nfan 1926	. . . . . . . . . . 11 ⊢ Ⅎ𝑡((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23		simp-5l 796	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝜑)
24		stoweidlem60.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 = (topGen‘ran (,))
25		stoweidlem60.4	. . . . . . . . . . . . . . . 16 ⊢ 𝑇 = ∪ 𝐽
26		stoweidlem60.5	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = (𝐽 Cn 𝐾)
27		stoweidlem60.15	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ 𝐶)
28	24, 25, 26, 27	fcnre 45671	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
29	28	ffvelcdmda 7080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
30	23, 29	sylancom 599	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
31		simp-5r 797	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ)
32	31	nnred 12248	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℝ)
33		simpllr 787	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℕ)
34	33	nnred 12248	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℝ)
35		1red 11209	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
36	34, 35	resubcld 11642	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝑛 − 1) ∈ ℝ)
37	23, 4	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
38	36, 37	remulcld 11239	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39		simpllr 787	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
40	39	r19.21bi 3263	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < 𝑚)
41		simplr 780	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43		simpl1 1208	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44		simpl2 1209	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
45	43, 44, 8	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46		1red 11209	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47		simpl3 1210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
48	47	nnred 12248	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
49	45, 46, 48	ltaddsubd 11814	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (((𝑚 / 𝐸) + 1) < 𝑛 ↔ (𝑚 / 𝐸) < (𝑛 − 1)))
50	42, 49	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) < (𝑛 − 1))
51	1	3ad2ant2 1150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
52	51	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
53	48, 46	resubcld 11642	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
54	4	3ad2ant1 1149	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
55	54	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
56	3	rpgt0d 13063	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 𝐸)
57	43, 56	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58		ltdivmul2 12092	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ℝ ∧ (𝑛 − 1) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
59	52, 53, 55, 57, 58	syl112anc 1399	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
60	50, 59	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 < ((𝑛 − 1) · 𝐸))
61	23, 31, 33, 41, 60	syl31anc 1398	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 < ((𝑛 − 1) · 𝐸))
62	30, 32, 38, 40, 61	lttrd 11371	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
63	62	ex 417	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑡 ∈ 𝑇 → (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
64	22, 63	ralrimi 3269	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
65	64	ex 417	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) → (((𝑚 / 𝐸) + 1) < 𝑛 → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
66	65	reximdva 3184	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → (∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
67	13, 66	mpd 16	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
68		stoweidlem60.1	. . . . . . . 8 ⊢ Ⅎ𝑡𝐹
69		stoweidlem60.8	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Comp)
70		stoweidlem60.9	. . . . . . . 8 ⊢ (𝜑 → 𝑇 ≠ ∅)
71	68, 14, 24, 69, 25, 70, 26, 27	rfcnnnub 45682	. . . . . . 7 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
72	67, 71	r19.29a 3179	. . . . . 6 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
73		df-rex 3096	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
74	72, 73	sylib 221	. . . . 5 ⊢ (𝜑 → ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
75		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
76	14, 19	nfan 1926	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ)
77		stoweidlem60.6	. . . . . . . . . . 11 ⊢ 𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
78		stoweidlem60.7	. . . . . . . . . . 11 ⊢ 𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡 ∈ 𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹‘𝑡)})
79		eqid 2769	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} = {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)}
80		eqid 2769	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0...𝑛) ↦ {𝑦 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} ∣ (∀𝑡 ∈ (𝐷‘𝑗)(𝑦‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < (𝑦‘𝑡))}) = (𝑗 ∈ (0...𝑛) ↦ {𝑦 ∈ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} ∣ (∀𝑡 ∈ (𝐷‘𝑗)(𝑦‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < (𝑦‘𝑡))})
81	69	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Comp)
82		stoweidlem60.10	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
83	82	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ⊆ 𝐶)
84		stoweidlem60.11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
85	84	3adant1r 1194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
86		stoweidlem60.12	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
87	86	3adant1r 1194	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
88		stoweidlem60.13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
89	88	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
90		stoweidlem60.14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
91	90	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
92	27	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ 𝐶)
93	3	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
94		stoweidlem60.18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 < (1 / 3))
95	94	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 < (1 / 3))
96		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
97	68, 76, 24, 25, 26, 77, 78, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 96	stoweidlem59 46699	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))
98	97	adantrr 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))
99		19.42v 1980	. . . . . . . . 9 ⊢ (∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ↔ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))))
100	75, 98, 99	sylanbrc 594	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))))
101		3anass 1109	. . . . . . . . 9 ⊢ (((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) ↔ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))))
102	101	exbii 1875	. . . . . . . 8 ⊢ (∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) ↔ ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))))
103	100, 102	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))
104	103	ex 417	. . . . . 6 ⊢ (𝜑 → ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))))
105	104	eximdv 1944	. . . . 5 ⊢ (𝜑 → (∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) → ∃𝑛∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))))
106	74, 105	mpd 16	. . . 4 ⊢ (𝜑 → ∃𝑛∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))
107		simpl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝜑)
108		simpr1l 1247	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝑛 ∈ ℕ)
109		simpr2 1212	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝑥:(0...𝑛)⟶𝐴)
110		nfv 1941	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥:(0...𝑛)⟶𝐴
111	14, 19, 110	nf3an 1928	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112		simp2 1153	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113		simp3 1154	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114		simp1 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115	114, 84	syl3an1 1179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
116	114, 86	syl3an1 1179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	88	3ad2antl1 1202	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
118	3	3ad2ant1 1149	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ⁺)
119	118	rpred 13060	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
120	82	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
121	24, 25, 26, 120	fcnre 45671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
122	121	3ad2antl1 1202	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
123	111, 112, 113, 115, 116, 117, 119, 122	stoweidlem17 46657	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴)
124	107, 108, 109, 123	syl3anc 1396	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴)
125		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
126		nfv 1941	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
127		nfv 1941	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥:(0...𝑛)⟶𝐴
128		nfra1 3295	. . . . . . . . . 10 ⊢ Ⅎ𝑗∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
129	126, 127, 128	nf3an 1928	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))
130	125, 129	nfan 1926	. . . . . . . 8 ⊢ Ⅎ𝑗(𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))
131		nfra1 3295	. . . . . . . . . . 11 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)
132	19, 131	nfan 1926	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
133		nfcv 2931	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(0...𝑛)
134		nfra1 3295	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1)
135		nfra1 3295	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)
136		nfra1 3295	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)
137	134, 135, 136	nf3an 1928	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
138	133, 137	nfralw 3318	. . . . . . . . . 10 ⊢ Ⅎ𝑡∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
139	132, 110, 138	nf3an 1928	. . . . . . . . 9 ⊢ Ⅎ𝑡((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))
140	14, 139	nfan 1926	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))))
141		eqid 2769	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)}) = (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)})
142	69	uniexd 7741	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
143	25, 142	eqeltrid 2873	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ V)
144	143	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝑇 ∈ V)
145	28	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝐹:𝑇⟶ℝ)
146		stoweidlem60.16	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐹‘𝑡))
147	146	r19.21bi 3263	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
148	147	adantlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
149		simpr1r 1248	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
150	149	r19.21bi 3263	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
151	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝐸 ∈ ℝ⁺)
152	94	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → 𝐸 < (1 / 3))
153		simpll 778	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝜑)
154		simplr2 1233	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝑥:(0...𝑛)⟶𝐴)
155		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ (0...𝑛))
156		simp1 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → 𝜑)
157		ffvelcdm 7077	. . . . . . . . . . 11 ⊢ ((𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
158	157	3adant1 1146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
159	82	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗) ∈ 𝐶)
160	24, 25, 26, 159	fcnre 45671	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗):𝑇⟶ℝ)
161	156, 158, 160	syl2anc 595	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ)
162	153, 154, 155, 161	syl3anc 1396	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ)
163		simp1r3 1288	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))
164		r19.26-3 3132	. . . . . . . . . . 11 ⊢ (∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) ↔ (∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))
165	164	simp1bi 1161	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1))
166		simpl 487	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥‘𝑗)‘𝑡))
167	166	2ralimi 3141	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
168	163, 165, 167	3syl 19	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
169		simp2 1153	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑗 ∈ (0...𝑛))
170		simp3 1154	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
171		rspa 3260	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
172	171	r19.21bi 3263	. . . . . . . . 9 ⊢ (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑥‘𝑗)‘𝑡))
173	168, 169, 170, 172	syl21anc 850	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑥‘𝑗)‘𝑡))
174		simpr 489	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ((𝑥‘𝑗)‘𝑡) ≤ 1)
175	174	2ralimi 3141	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
176	163, 165, 175	3syl 19	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
177		rspa 3260	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
178	177	r19.21bi 3263	. . . . . . . . 9 ⊢ (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ 𝑇) → ((𝑥‘𝑗)‘𝑡) ≤ 1)
179	176, 169, 170, 178	syl21anc 850	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑥‘𝑗)‘𝑡) ≤ 1)
180		simp1r3 1288	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))
181	164	simp2bi 1162	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛))
182	180, 181	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛))
183		simp2 1153	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → 𝑗 ∈ (0...𝑛))
184		simp3 1154	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → 𝑡 ∈ (𝐷‘𝑗))
185		rspa 3260	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛))
186	185	r19.21bi 3263	. . . . . . . . 9 ⊢ (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛))
187	182, 183, 184, 186	syl21anc 850	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷‘𝑗)) → ((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛))
188		simp1r3 1288	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))
189	164	simp3bi 1163	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
190	188, 189	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
191		simp2 1153	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → 𝑗 ∈ (0...𝑛))
192		simp3 1154	. . . . . . . . 9 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → 𝑡 ∈ (𝐵‘𝑗))
193		rspa 3260	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
194	193	r19.21bi 3263	. . . . . . . . 9 ⊢ (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐵‘𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
195	190, 191, 192, 194	syl21anc 850	. . . . . . . 8 ⊢ (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵‘𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))
196	68, 130, 140, 77, 78, 141, 108, 144, 145, 148, 150, 151, 152, 162, 173, 179, 187, 195	stoweidlem34 46674	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))))
197		nfmpt1 5214	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
198	197	nfeq2 2948	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
199		fveq1 6881	. . . . . . . . . . . . 13 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (𝑔‘𝑡) = ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))
200	199	breq1d 5123	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ↔ ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸)))
201	199	breq2d 5125	. . . . . . . . . . . 12 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡) ↔ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))
202	200, 201	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)) ↔ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡))))
203	202	anbi2d 641	. . . . . . . . . 10 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) ↔ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))))
204	203	rexbidv 3195	. . . . . . . . 9 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) ↔ ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))))
205	198, 204	ralbid 3284	. . . . . . . 8 ⊢ (𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) → (∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) ↔ ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))))
206	205	rspcev 3590	. . . . . . 7 ⊢ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))‘𝑡)))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))
207	124, 196, 206	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))
208	207	ex 417	. . . . 5 ⊢ (𝜑 → (((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))))
209	208	2eximdv 1946	. . . 4 ⊢ (𝜑 → (∃𝑛∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡))) → ∃𝑛∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))))
210	106, 209	mpd 16	. . 3 ⊢ (𝜑 → ∃𝑛∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))
211		idd 25	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))))
212	211	exlimdv 1960	. . 3 ⊢ (𝜑 → (∃𝑛∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))))
213	210, 212	mpd 16	. 2 ⊢ (𝜑 → ∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))
214		idd 25	. . 3 ⊢ (𝜑 → (∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))))
215	214	exlimdv 1960	. 2 ⊢ (𝜑 → (∃𝑥∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))) → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡)))))
216	213, 215	mpd 16	1 ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 ∀𝑡 ∈ 𝑇 ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹‘𝑡) ∧ (𝐹‘𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔‘𝑡))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∃wex 1806 Ⅎwnf 1810 ∈ wcel 2149 Ⅎwnfc 2916 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3423 Vcvv 3463 ⊆ wss 3913 ∅c0 4294 ∪ cuni 4876 class class class wbr 5113 ↦ cmpt 5196 ran crn 5663 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 ≤ cle 11244 − cmin 11441 / cdiv 11871 ℕcn 12233 3c3 12296 4c4 12297 ℝ⁺crp 13016 (,)cioo 13372 ...cfz 13535 Σcsu 15737 topGenctg 17490 Cn ccn 23350 Compccmp 23512

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-inf2 9610 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 ax-pre-sup 11178

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-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 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-se 5616 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-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 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-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-cn 23353 df-cnp 23354 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-xms 24446 df-ms 24447 df-tms 24448

This theorem is referenced by: stoweidlem61 46701

W3C validator