stoweidlem60 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem60 44711

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 12215	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
2	1	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3		stoweidlem60.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 13012	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	4	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
6	3	rpne0d 13017	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≠ 0)
7	6	adantr 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ≠ 0)
8	2, 5, 7	redivcld 12038	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9		1red 11211	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ)
10	8, 9	readdcld 11239	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
11	10	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12		arch 12465	. . . . . . . . 9 ⊢ (((𝑚 / 𝐸) + 1) ∈ ℝ → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛)
13	11, 12	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛)
14		stoweidlem60.2	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝜑
15		nfv 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
16	14, 15	nfan 1903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
17		nfra1 3282	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚
18	16, 17	nfan 1903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
19		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡 𝑛 ∈ ℕ
20	18, 19	nfan 1903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡(((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21		nfv 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑚 / 𝐸) + 1) < 𝑛
22	20, 21	nfan 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑡((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23		simp-5l 784	. . . . . . . . . . . . . 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 43642	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
29	28	ffvelcdmda 7082	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
30	23, 29	sylancom 589	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
31		simp-5r 785	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ)
32	31	nnred 12223	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℝ)
33		simpllr 775	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℕ)
34	33	nnred 12223	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℝ)
35		1red 11211	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
36	34, 35	resubcld 11638	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝑛 − 1) ∈ ℝ)
37	23, 4	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝐸 ∈ ℝ)
38	36, 37	remulcld 11240	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39		simpllr 775	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
40	39	r19.21bi 3249	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < 𝑚)
41		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43		simpl1 1192	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44		simpl2 1193	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
45	43, 44, 8	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46		1red 11211	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47		simpl3 1194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
48	47	nnred 12223	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
49	45, 46, 48	ltaddsubd 11810	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (((𝑚 / 𝐸) + 1) < 𝑛 ↔ (𝑚 / 𝐸) < (𝑛 − 1)))
50	42, 49	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) < (𝑛 − 1))
51	1	3ad2ant2 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
52	51	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
53	48, 46	resubcld 11638	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
54	4	3ad2ant1 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
55	54	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
56	3	rpgt0d 13015	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 𝐸)
57	43, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58		ltdivmul2 12087	. . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
60	50, 59	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 < ((𝑛 − 1) · 𝐸))
61	23, 31, 33, 41, 60	syl31anc 1374	. . . . . . . . . . . . 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 414	. . . . . . . . . . 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 3255	. . . . . . . . . 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 414	. . . . . . . . 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 3169	. . . . . . . 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 15	. . . . . . 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 43653	. . . . . . 7 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
72	67, 71	r19.29a 3163	. . . . . 6 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
73		df-rex 3072	. . . . . 6 ⊢ (∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
74	72, 73	sylib 217	. . . . 5 ⊢ (𝜑 → ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
75		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
76	14, 19	nfan 1903	. . . . . . . . . . 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 2733	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} = {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)}
80		eqid 2733	. . . . . . . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Comp)
82		stoweidlem60.10	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
83	82	adantr 482	. . . . . . . . . . 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 1178	. . . . . . . . . . 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 1178	. . . . . . . . . . 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 714	. . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
92	27	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ 𝐶)
93	3	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
94		stoweidlem60.18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 < (1 / 3))
95	94	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 < (1 / 3))
96		simpr 486	. . . . . . . . . . 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 44710	. . . . . . . . . 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 716	. . . . . . . . 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 1958	. . . . . . . . 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 584	. . . . . . . 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 1096	. . . . . . . . 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 1851	. . . . . . . 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 233	. . . . . . 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 414	. . . . . 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 1921	. . . . 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 15	. . . 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 484	. . . . . . . 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 1231	. . . . . . . 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 1196	. . . . . . . 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 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥:(0...𝑛)⟶𝐴
111	14, 19, 110	nf3an 1905	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112		simp2 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113		simp3 1139	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114		simp1 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115	114, 84	syl3an1 1164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
116	114, 86	syl3an1 1164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	88	3ad2antl1 1186	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
118	3	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ⁺)
119	118	rpred 13012	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
120	82	sselda 3981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
121	24, 25, 26, 120	fcnre 43642	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
122	121	3ad2antl1 1186	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
123	111, 112, 113, 115, 116, 117, 119, 122	stoweidlem17 44668	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴)
124	107, 108, 109, 123	syl3anc 1372	. . . . . . 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 1918	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
126		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
127		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥:(0...𝑛)⟶𝐴
128		nfra1 3282	. . . . . . . . . 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 1905	. . . . . . . . 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 1903	. . . . . . . 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 3282	. . . . . . . . . . 11 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)
132	19, 131	nfan 1903	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
133		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(0...𝑛)
134		nfra1 3282	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1)
135		nfra1 3282	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)
136		nfra1 3282	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)
137	134, 135, 136	nf3an 1905	. . . . . . . . . . 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 3309	. . . . . . . . . 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 1905	. . . . . . . . 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 1903	. . . . . . . 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 2733	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)}) = (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)})
142	69	uniexd 7727	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
143	25, 142	eqeltrid 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ V)
144	143	adantr 482	. . . . . . . 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 482	. . . . . . . 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 3249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
148	147	adantlr 714	. . . . . . . 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 1232	. . . . . . . . 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 3249	. . . . . . . 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 482	. . . . . . . 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 482	. . . . . . . 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 766	. . . . . . . . 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 1217	. . . . . . . . 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 486	. . . . . . . . 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 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → 𝜑)
157		ffvelcdm 7079	. . . . . . . . . . 11 ⊢ ((𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
158	157	3adant1 1131	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
159	82	sselda 3981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗) ∈ 𝐶)
160	24, 25, 26, 159	fcnre 43642	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗):𝑇⟶ℝ)
161	156, 158, 160	syl2anc 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ)
162	153, 154, 155, 161	syl3anc 1372	. . . . . . . 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 1272	. . . . . . . . . 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 3113	. . . . . . . . . . 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 1146	. . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥‘𝑗)‘𝑡))
167	166	2ralimi 3124	. . . . . . . . . 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 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 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
169		simp2 1138	. . . . . . . . 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 1139	. . . . . . . . 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 3246	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
172	171	r19.21bi 3249	. . . . . . . . 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 837	. . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ((𝑥‘𝑗)‘𝑡) ≤ 1)
175	174	2ralimi 3124	. . . . . . . . . 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 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 ∈ 𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
177		rspa 3246	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
178	177	r19.21bi 3249	. . . . . . . . 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 837	. . . . . . . 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 1272	. . . . . . . . . 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 1147	. . . . . . . . . 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 17	. . . . . . . . 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 1138	. . . . . . . . 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 1139	. . . . . . . . 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 3246	. . . . . . . . . 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 3249	. . . . . . . . 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 837	. . . . . . . 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 1272	. . . . . . . . . 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 1148	. . . . . . . . . 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 17	. . . . . . . . 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 1138	. . . . . . . . 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 1139	. . . . . . . . 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 3246	. . . . . . . . . 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 3249	. . . . . . . . 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 837	. . . . . . . 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 44685	. . . . . . 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 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
198	197	nfeq2 2921	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
199		fveq1 6887	. . . . . . . . . . . . 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 5157	. . . . . . . . . . . 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 5159	. . . . . . . . . . . 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 632	. . . . . . . . . . 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 630	. . . . . . . . . 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 3179	. . . . . . . . 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 3271	. . . . . . . 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 3612	. . . . . . 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 585	. . . . . 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 414	. . . . 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 1923	. . . 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 15	. . 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 24	. . . 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 1937	. . 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 15	. 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 24	. . 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 1937	. 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 15	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 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3947 ∅c0 4321 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 3c3 12264 4c4 12265 ℝ⁺crp 12970 (,)cioo 13320 ...cfz 13480 Σcsu 15628 topGenctg 17379 Cn ccn 22710 Compccmp 22872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-cn 22713 df-cnp 22714 df-cmp 22873 df-tx 23048 df-hmeo 23241 df-xms 23808 df-ms 23809 df-tms 23810

This theorem is referenced by: stoweidlem61 44712

W3C validator