stoweidlem60 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem60 44291

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 12160	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
2	1	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3		stoweidlem60.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 12957	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
6	3	rpne0d 12962	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≠ 0)
7	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ≠ 0)
8	2, 5, 7	redivcld 11983	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9		1red 11156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ)
10	8, 9	readdcld 11184	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
11	10	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12		arch 12410	. . . . . . . . 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 1917	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
16	14, 15	nfan 1902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
17		nfra1 3267	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚
18	16, 17	nfan 1902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
19		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡 𝑛 ∈ ℕ
20	18, 19	nfan 1902	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡(((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑚 / 𝐸) + 1) < 𝑛
22	20, 21	nfan 1902	. . . . . . . . . . 11 ⊢ Ⅎ𝑡((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23		simp-5l 783	. . . . . . . . . . . . . 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 43220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
29	28	ffvelcdmda 7035	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
30	23, 29	sylancom 588	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
31		simp-5r 784	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ)
32	31	nnred 12168	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℝ)
33		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℕ)
34	33	nnred 12168	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℝ)
35		1red 11156	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
36	34, 35	resubcld 11583	. . . . . . . . . . . . . 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 11185	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39		simpllr 774	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
40	39	r19.21bi 3234	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < 𝑚)
41		simplr 767	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43		simpl1 1191	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44		simpl2 1192	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
45	43, 44, 8	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46		1red 11156	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47		simpl3 1193	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
48	47	nnred 12168	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
49	45, 46, 48	ltaddsubd 11755	. . . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
52	51	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
53	48, 46	resubcld 11583	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
54	4	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
55	54	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
56	3	rpgt0d 12960	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 𝐸)
57	43, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58		ltdivmul2 12032	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 11316	. . . . . . . . . . . 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 413	. . . . . . . . . . 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 3240	. . . . . . . . . 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 413	. . . . . . . . 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 3165	. . . . . . . 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 43231	. . . . . . 7 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
72	67, 71	r19.29a 3159	. . . . . 6 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
73		df-rex 3074	. . . . . 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 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
76	14, 19	nfan 1902	. . . . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} = {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)}
80		eqid 2736	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Comp)
82		stoweidlem60.10	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
83	82	adantr 481	. . . . . . . . . . 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 1177	. . . . . . . . . . 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 1177	. . . . . . . . . . 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 713	. . . . . . . . . . 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 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
92	27	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ 𝐶)
93	3	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
94		stoweidlem60.18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 < (1 / 3))
95	94	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 < (1 / 3))
96		simpr 485	. . . . . . . . . . 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 44290	. . . . . . . . . 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 715	. . . . . . . . 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 1957	. . . . . . . . 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 583	. . . . . . . 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 1095	. . . . . . . . 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 1850	. . . . . . . 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 413	. . . . . 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 1920	. . . . 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 483	. . . . . . . 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 1230	. . . . . . . 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 1195	. . . . . . . 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 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥:(0...𝑛)⟶𝐴
111	14, 19, 110	nf3an 1904	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112		simp2 1137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114		simp1 1136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115	114, 84	syl3an1 1163	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
116	114, 86	syl3an1 1163	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	88	3ad2antl1 1185	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
118	3	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ⁺)
119	118	rpred 12957	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
120	82	sselda 3944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
121	24, 25, 26, 120	fcnre 43220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
122	121	3ad2antl1 1185	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
123	111, 112, 113, 115, 116, 117, 119, 122	stoweidlem17 44248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴)
124	107, 108, 109, 123	syl3anc 1371	. . . . . . 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 1917	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
126		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
127		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥:(0...𝑛)⟶𝐴
128		nfra1 3267	. . . . . . . . . 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 1904	. . . . . . . . 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 1902	. . . . . . . 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 3267	. . . . . . . . . . 11 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)
132	19, 131	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
133		nfcv 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(0...𝑛)
134		nfra1 3267	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1)
135		nfra1 3267	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)
136		nfra1 3267	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)
137	134, 135, 136	nf3an 1904	. . . . . . . . . . 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 3294	. . . . . . . . . 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 1904	. . . . . . . . 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 1902	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)}) = (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)})
142	69	uniexd 7679	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
143	25, 142	eqeltrid 2842	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ V)
144	143	adantr 481	. . . . . . . 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 481	. . . . . . . 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 3234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
148	147	adantlr 713	. . . . . . . 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 1231	. . . . . . . . 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 3234	. . . . . . . 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 481	. . . . . . . 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 481	. . . . . . . 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 765	. . . . . . . . 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 1216	. . . . . . . . 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 485	. . . . . . . . 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 1136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → 𝜑)
157		ffvelcdm 7032	. . . . . . . . . . 11 ⊢ ((𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
158	157	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
159	82	sselda 3944	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗) ∈ 𝐶)
160	24, 25, 26, 159	fcnre 43220	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗):𝑇⟶ℝ)
161	156, 158, 160	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ)
162	153, 154, 155, 161	syl3anc 1371	. . . . . . . 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 1271	. . . . . . . . . 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 3115	. . . . . . . . . . 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 1145	. . . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥‘𝑗)‘𝑡))
167	166	2ralimi 3126	. . . . . . . . . 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 1137	. . . . . . . . 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 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) → 𝑡 ∈ 𝑇)
171		rspa 3231	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
172	171	r19.21bi 3234	. . . . . . . . 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 836	. . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ((𝑥‘𝑗)‘𝑡) ≤ 1)
175	174	2ralimi 3126	. . . . . . . . . 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 3231	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
178	177	r19.21bi 3234	. . . . . . . . 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 836	. . . . . . . 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 1271	. . . . . . . . . 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 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 ∈ (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 1137	. . . . . . . . 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 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)) → 𝑡 ∈ (𝐷‘𝑗))
185		rspa 3231	. . . . . . . . . 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 3234	. . . . . . . . 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 836	. . . . . . . 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 1271	. . . . . . . . . 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 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 ∈ (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 1137	. . . . . . . . 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 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)) → 𝑡 ∈ (𝐵‘𝑗))
193		rspa 3231	. . . . . . . . . 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 3234	. . . . . . . . 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 836	. . . . . . . 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 44265	. . . . . . 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 5213	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
198	197	nfeq2 2924	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
199		fveq1 6841	. . . . . . . . . . . . 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 5115	. . . . . . . . . . . 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 5117	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 629	. . . . . . . . . 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 3175	. . . . . . . . 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 3256	. . . . . . . 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 3581	. . . . . . 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 584	. . . . . 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 413	. . . . 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 1922	. . . 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 1936	. . 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 1936	. 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 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2887 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ∪ cuni 4865 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 3c3 12209 4c4 12210 ℝ⁺crp 12915 (,)cioo 13264 ...cfz 13424 Σcsu 15570 topGenctg 17319 Cn ccn 22575 Compccmp 22737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-cn 22578 df-cnp 22579 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675

This theorem is referenced by: stoweidlem61 44292

W3C validator