stoweidlem60 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem60 46851

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 12259	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
2	1	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3		stoweidlem60.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
4	3	rpred 13080	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ∈ ℝ)
5	4	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
6	3	rpne0d 13085	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐸 ≠ 0)
7	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐸 ≠ 0)
8	2, 5, 7	redivcld 12062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9		1red 11228	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ)
10	8, 9	readdcld 11257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
11	10	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12		arch 12520	. . . . . . . . 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 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
16	14, 15	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
17		nfra1 3291	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚
18	16, 17	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
19		nfv 1947	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡 𝑛 ∈ ℕ
20	18, 19	nfan 1932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡(((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡((𝑚 / 𝐸) + 1) < 𝑛
22	20, 21	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑡((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23		simp-5l 797	. . . . . . . . . . . . . 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 45822	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
29	28	ffvelcdmda 7083	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
30	23, 29	sylancom 600	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ)
31		simp-5r 798	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℕ)
32	31	nnred 12267	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑚 ∈ ℝ)
33		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℕ)
34	33	nnred 12267	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 𝑛 ∈ ℝ)
35		1red 11228	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
36	34, 35	resubcld 11661	. . . . . . . . . . . . . 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 11258	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39		simpllr 788	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
40	39	r19.21bi 3259	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) < 𝑚)
41		simplr 781	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡 ∈ 𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43		simpl1 1210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44		simpl2 1211	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
45	43, 44, 8	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46		1red 11228	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47		simpl3 1212	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
48	47	nnred 12267	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
49	45, 46, 48	ltaddsubd 11833	. . . . . . . . . . . . . . . 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 1152	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
52	51	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
53	48, 46	resubcld 11661	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
54	4	3ad2ant1 1151	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
55	54	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
56	3	rpgt0d 13083	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 < 𝐸)
57	43, 56	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58		ltdivmul2 12111	. . . . . . . . . . . . . . . 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 1401	. . . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . 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 11390	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 3265	. . . . . . . . . 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 418	. . . . . . . . 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 3180	. . . . . . . 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 45833	. . . . . . 7 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑚)
72	67, 71	r19.29a 3175	. . . . . 6 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
73		df-rex 3092	. . . . . 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 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)))
76	14, 19	nfan 1932	. . . . . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)} = {𝑦 ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)}
80		eqid 2765	. . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐽 ∈ Comp)
82		stoweidlem60.10	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
83	82	adantr 486	. . . . . . . . . . 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 1196	. . . . . . . . . . 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 1196	. . . . . . . . . . 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 728	. . . . . . . . . . 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 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
92	27	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐹 ∈ 𝐶)
93	3	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ⁺)
94		stoweidlem60.18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 < (1 / 3))
95	94	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐸 < (1 / 3))
96		simpr 490	. . . . . . . . . . 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 46850	. . . . . . . . . 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 730	. . . . . . . . 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 1986	. . . . . . . . 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 595	. . . . . . . 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 1111	. . . . . . . . 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 1881	. . . . . . . 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 418	. . . . . 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 1950	. . . . 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 488	. . . . . . . 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 1249	. . . . . . . 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 1214	. . . . . . . 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 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥:(0...𝑛)⟶𝐴
111	14, 19, 110	nf3an 1934	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112		simp2 1155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114		simp1 1154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115	114, 84	syl3an1 1181	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
116	114, 86	syl3an1 1181	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	88	3ad2antl1 1204	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
118	3	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ⁺)
119	118	rpred 13080	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
120	82	sselda 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
121	24, 25, 26, 120	fcnre 45822	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
122	121	3ad2antl1 1204	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
123	111, 112, 113, 115, 116, 117, 119, 122	stoweidlem17 46808	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡))) ∈ 𝐴)
124	107, 108, 109, 123	syl3anc 1398	. . . . . . 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 1947	. . . . . . . . 9 ⊢ Ⅎ𝑗𝜑
126		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑗(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
127		nfv 1947	. . . . . . . . . 10 ⊢ Ⅎ𝑗 𝑥:(0...𝑛)⟶𝐴
128		nfra1 3291	. . . . . . . . . 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 1934	. . . . . . . . 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 1932	. . . . . . . 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 3291	. . . . . . . . . . 11 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸)
132	19, 131	nfan 1932	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑛 ∈ ℕ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < ((𝑛 − 1) · 𝐸))
133		nfcv 2927	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(0...𝑛)
134		nfra1 3291	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1)
135		nfra1 3291	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐷‘𝑗)((𝑥‘𝑗)‘𝑡) < (𝐸 / 𝑛)
136		nfra1 3291	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝐵‘𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥‘𝑗)‘𝑡)
137	134, 135, 136	nf3an 1934	. . . . . . . . . . 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 3314	. . . . . . . . . 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 1934	. . . . . . . . 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 1932	. . . . . . . 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 2765	. . . . . . . 8 ⊢ (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)}) = (𝑡 ∈ 𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷‘𝑗)})
142	69	uniexd 7750	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
143	25, 142	eqeltrid 2869	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ V)
144	143	adantr 486	. . . . . . . 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 486	. . . . . . . 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 3259	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐹‘𝑡))
148	147	adantlr 728	. . . . . . . 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 1250	. . . . . . . . 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 3259	. . . . . . . 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 486	. . . . . . . 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 486	. . . . . . . 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 779	. . . . . . . . 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 1235	. . . . . . . . 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 490	. . . . . . . . 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 1154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → 𝜑)
157		ffvelcdm 7080	. . . . . . . . . . 11 ⊢ ((𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
158	157	3adant1 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗) ∈ 𝐴)
159	82	sselda 3938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗) ∈ 𝐶)
160	24, 25, 26, 159	fcnre 45822	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥‘𝑗) ∈ 𝐴) → (𝑥‘𝑗):𝑇⟶ℝ)
161	156, 158, 160	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ 𝑗 ∈ (0...𝑛)) → (𝑥‘𝑗):𝑇⟶ℝ)
162	153, 154, 155, 161	syl3anc 1398	. . . . . . . 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 1290	. . . . . . . . . 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 3128	. . . . . . . . . . 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 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 ∈ 𝑇 (0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1))
166		simpl 488	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥‘𝑗)‘𝑡))
167	166	2ralimi 3137	. . . . . . . . . 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 1155	. . . . . . . . 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 1156	. . . . . . . . 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 3256	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 0 ≤ ((𝑥‘𝑗)‘𝑡))
172	171	r19.21bi 3259	. . . . . . . . 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 851	. . . . . . . 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 490	. . . . . . . . . . 11 ⊢ ((0 ≤ ((𝑥‘𝑗)‘𝑡) ∧ ((𝑥‘𝑗)‘𝑡) ≤ 1) → ((𝑥‘𝑗)‘𝑡) ≤ 1)
175	174	2ralimi 3137	. . . . . . . . . 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 3256	. . . . . . . . . 10 ⊢ ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ 𝑇 ((𝑥‘𝑗)‘𝑡) ≤ 1)
178	177	r19.21bi 3259	. . . . . . . . 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 851	. . . . . . . 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 1290	. . . . . . . . . 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 1164	. . . . . . . . . 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 1155	. . . . . . . . 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 1156	. . . . . . . . 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 3256	. . . . . . . . . 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 3259	. . . . . . . . 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 851	. . . . . . . 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 1290	. . . . . . . . . 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 1165	. . . . . . . . . 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 1155	. . . . . . . . 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 1156	. . . . . . . . 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 3256	. . . . . . . . . 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 3259	. . . . . . . . 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 851	. . . . . . . 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 46825	. . . . . . 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 5212	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
198	197	nfeq2 2944	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑔 = (𝑡 ∈ 𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥‘𝑖)‘𝑡)))
199		fveq1 6884	. . . . . . . . . . . . 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 5121	. . . . . . . . . . . 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 5123	. . . . . . . . . . . 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 644	. . . . . . . . . . 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 642	. . . . . . . . . 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 3191	. . . . . . . . 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 3280	. . . . . . . 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 3583	. . . . . . 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 596	. . . . . 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 418	. . . . 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 1952	. . . 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 1966	. . 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 1966	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∃wex 1812 Ⅎwnf 1816 ∈ wcel 2146 Ⅎwnfc 2912 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 {crab 3418 Vcvv 3457 ⊆ wss 3906 ∅c0 4286 ∪ cuni 4874 class class class wbr 5111 ↦ cmpt 5194 ran crn 5664 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ℝcr 11118 0cc0 11119 1c1 11120 + caddc 11122 · cmul 11124 < clt 11262 ≤ cle 11263 − cmin 11460 / cdiv 11890 ℕcn 12252 3c3 12315 4c4 12316 ℝ⁺crp 13036 (,)cioo 13392 ...cfz 13555 Σcsu 15765 topGenctg 17516 Cn ccn 23435 Compccmp 23597

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 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 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-pm 8833 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-fi 9378 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-q 12993 df-rp 13037 df-xneg 13157 df-xadd 13158 df-xmul 13159 df-ioo 13396 df-ioc 13397 df-ico 13398 df-icc 13399 df-fz 13556 df-fzo 13704 df-fl 13847 df-seq 14060 df-exp 14120 df-hash 14389 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-clim 15567 df-rlim 15568 df-sum 15766 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-starv 17351 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-unif 17359 df-hom 17360 df-cco 17361 df-rest 17501 df-topn 17502 df-0g 17520 df-gsum 17521 df-topgen 17522 df-pt 17523 df-prds 17526 df-xrs 17582 df-qtop 17587 df-imas 17588 df-xps 17590 df-mre 17664 df-mrc 17665 df-acs 17667 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-submnd 18883 df-mulg 19182 df-cntz 19435 df-cmn 19900 df-psmet 21568 df-xmet 21569 df-met 21570 df-bl 21571 df-mopn 21572 df-cnfld 21577 df-top 23105 df-topon 23122 df-topsp 23144 df-bases 23157 df-cld 23230 df-cn 23438 df-cnp 23439 df-cmp 23598 df-tx 23774 df-hmeo 23967 df-xms 24532 df-ms 24533 df-tms 24534

This theorem is used by: stoweidlem61 46852

W3C validator