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Theorem stoweidlem60 39581
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 𝑡𝐹
stoweidlem60.2 𝑡𝜑
stoweidlem60.3 𝐾 = (topGen‘ran (,))
stoweidlem60.4 𝑇 = 𝐽
stoweidlem60.5 𝐶 = (𝐽 Cn 𝐾)
stoweidlem60.6 𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
stoweidlem60.7 𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
stoweidlem60.8 (𝜑𝐽 ∈ Comp)
stoweidlem60.9 (𝜑𝑇 ≠ ∅)
stoweidlem60.10 (𝜑𝐴𝐶)
stoweidlem60.11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
stoweidlem60.12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem60.13 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
stoweidlem60.14 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
stoweidlem60.15 (𝜑𝐹𝐶)
stoweidlem60.16 (𝜑 → ∀𝑡𝑇 0 ≤ (𝐹𝑡))
stoweidlem60.17 (𝜑𝐸 ∈ ℝ+)
stoweidlem60.18 (𝜑𝐸 < (1 / 3))
Assertion
Ref Expression
stoweidlem60 (𝜑 → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
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.
StepHypRef Expression
1 nnre 10971 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
21adantl 482 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3 stoweidlem60.17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ ℝ+)
43rpred 11816 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ℝ)
54adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
63rpne0d 11821 . . . . . . . . . . . . 13 (𝜑𝐸 ≠ 0)
76adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ≠ 0)
82, 5, 7redivcld 10797 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9 1red 9999 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℝ)
108, 9readdcld 10013 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
1110adantr 481 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12 arch 11233 . . . . . . . . 9 (((𝑚 / 𝐸) + 1) ∈ ℝ → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛)
1311, 12syl 17 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛)
14 stoweidlem60.2 . . . . . . . . . . . . . . 15 𝑡𝜑
15 nfv 1840 . . . . . . . . . . . . . . 15 𝑡 𝑚 ∈ ℕ
1614, 15nfan 1825 . . . . . . . . . . . . . 14 𝑡(𝜑𝑚 ∈ ℕ)
17 nfra1 2936 . . . . . . . . . . . . . 14 𝑡𝑡𝑇 (𝐹𝑡) < 𝑚
1816, 17nfan 1825 . . . . . . . . . . . . 13 𝑡((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
19 nfv 1840 . . . . . . . . . . . . 13 𝑡 𝑛 ∈ ℕ
2018, 19nfan 1825 . . . . . . . . . . . 12 𝑡(((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21 nfv 1840 . . . . . . . . . . . 12 𝑡((𝑚 / 𝐸) + 1) < 𝑛
2220, 21nfan 1825 . . . . . . . . . . 11 𝑡((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23 simp-5l 807 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 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 (𝜑𝐹𝐶)
2824, 25, 26, 27fcnre 38664 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑇⟶ℝ)
2928ffvelrnda 6315 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
3023, 29sylancom 700 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
31 simp-5r 808 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ)
3231nnred 10979 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℝ)
33 simpllr 798 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℕ)
3433nnred 10979 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℝ)
35 1red 9999 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 1 ∈ ℝ)
3634, 35resubcld 10402 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝑛 − 1) ∈ ℝ)
3723, 4syl 17 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝐸 ∈ ℝ)
3836, 37remulcld 10014 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39 simpllr 798 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
4039r19.21bi 2927 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < 𝑚)
41 simplr 791 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43 simpl1 1062 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44 simpl2 1063 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
4543, 44, 8syl2anc 692 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46 1red 9999 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47 simpl3 1064 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
4847nnred 10979 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
4945, 46, 48ltaddsubd 10571 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (((𝑚 / 𝐸) + 1) < 𝑛 ↔ (𝑚 / 𝐸) < (𝑛 − 1)))
5042, 49mpbid 222 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) < (𝑛 − 1))
5113ad2ant2 1081 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
5251adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
5348, 46resubcld 10402 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
5443ad2ant1 1080 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
5554adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
563rpgt0d 11819 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝐸)
5743, 56syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58 ltdivmul2 10844 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℝ ∧ (𝑛 − 1) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
5952, 53, 55, 57, 58syl112anc 1327 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
6050, 59mpbid 222 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 < ((𝑛 − 1) · 𝐸))
6123, 31, 33, 41, 60syl31anc 1326 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 < ((𝑛 − 1) · 𝐸))
6230, 32, 38, 40, 61lttrd 10142 . . . . . . . . . . . 12 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
6362ex 450 . . . . . . . . . . 11 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑡𝑇 → (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6422, 63ralrimi 2951 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
6564ex 450 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) → (((𝑚 / 𝐸) + 1) < 𝑛 → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6665reximdva 3011 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → (∃𝑛 ∈ ℕ ((𝑚 / 𝐸) + 1) < 𝑛 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6713, 66mpd 15 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
68 stoweidlem60.1 . . . . . . . 8 𝑡𝐹
69 stoweidlem60.8 . . . . . . . 8 (𝜑𝐽 ∈ Comp)
70 stoweidlem60.9 . . . . . . . 8 (𝜑𝑇 ≠ ∅)
7168, 14, 24, 69, 25, 70, 26, 27rfcnnnub 38675 . . . . . . 7 (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
7267, 71r19.29a 3071 . . . . . 6 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
73 df-rex 2913 . . . . . 6 (∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
7472, 73sylib 208 . . . . 5 (𝜑 → ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
75 simpr 477 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
7614, 19nfan 1825 . . . . . . . . . . 11 𝑡(𝜑𝑛 ∈ ℕ)
77 stoweidlem60.6 . . . . . . . . . . 11 𝐷 = (𝑗 ∈ (0...𝑛) ↦ {𝑡𝑇 ∣ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)})
78 stoweidlem60.7 . . . . . . . . . . 11 𝐵 = (𝑗 ∈ (0...𝑛) ↦ {𝑡𝑇 ∣ ((𝑗 + (1 / 3)) · 𝐸) ≤ (𝐹𝑡)})
79 eqid 2621 . . . . . . . . . . 11 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
80 eqid 2621 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑛) ↦ {𝑦 ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < (𝑦𝑡))}) = (𝑗 ∈ (0...𝑛) ↦ {𝑦 ∈ {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} ∣ (∀𝑡 ∈ (𝐷𝑗)(𝑦𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < (𝑦𝑡))})
8169adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐽 ∈ Comp)
82 stoweidlem60.10 . . . . . . . . . . . 12 (𝜑𝐴𝐶)
8382adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴𝐶)
84 stoweidlem60.11 . . . . . . . . . . . 12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
85843adant1r 1316 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
86 stoweidlem60.12 . . . . . . . . . . . 12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
87863adant1r 1316 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
88 stoweidlem60.13 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
8988adantlr 750 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
90 stoweidlem60.14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9190adantlr 750 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9227adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹𝐶)
933adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐸 ∈ ℝ+)
94 stoweidlem60.18 . . . . . . . . . . . 12 (𝜑𝐸 < (1 / 3))
9594adantr 481 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐸 < (1 / 3))
96 simpr 477 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9768, 76, 24, 25, 26, 77, 78, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 96stoweidlem59 39580 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))))
9897adantrr 752 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))))
99 19.42v 1915 . . . . . . . . 9 (∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ↔ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ ∃𝑥(𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))))
10075, 98, 99sylanbrc 697 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))))
101 3anass 1040 . . . . . . . . 9 (((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))) ↔ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))))
102101exbii 1771 . . . . . . . 8 (∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))) ↔ ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ (𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))))
103100, 102sylibr 224 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))))
104103ex 450 . . . . . 6 (𝜑 → ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) → ∃𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))))
105104eximdv 1843 . . . . 5 (𝜑 → (∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) → ∃𝑛𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))))
10674, 105mpd 15 . . . 4 (𝜑 → ∃𝑛𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))))
107 simpl 473 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝜑)
108 simpr1l 1116 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝑛 ∈ ℕ)
109 simpr2 1066 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝑥:(0...𝑛)⟶𝐴)
110 nfv 1840 . . . . . . . . . 10 𝑡 𝑥:(0...𝑛)⟶𝐴
11114, 19, 110nf3an 1828 . . . . . . . . 9 𝑡(𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112 simp2 1060 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113 simp3 1061 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114 simp1 1059 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115114, 84syl3an1 1356 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116114, 86syl3an1 1356 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
117883ad2antl1 1221 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
11833ad2ant1 1080 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ+)
119118rpred 11816 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
12082sselda 3583 . . . . . . . . . . 11 ((𝜑𝑓𝐴) → 𝑓𝐶)
12124, 25, 26, 120fcnre 38664 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1221213ad2antl1 1221 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
123111, 112, 113, 115, 116, 117, 119, 122stoweidlem17 39538 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) ∈ 𝐴)
124107, 108, 109, 123syl3anc 1323 . . . . . . 7 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) ∈ 𝐴)
125 nfv 1840 . . . . . . . . 9 𝑗𝜑
126 nfv 1840 . . . . . . . . . 10 𝑗(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
127 nfv 1840 . . . . . . . . . 10 𝑗 𝑥:(0...𝑛)⟶𝐴
128 nfra1 2936 . . . . . . . . . 10 𝑗𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
129126, 127, 128nf3an 1828 . . . . . . . . 9 𝑗((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))
130125, 129nfan 1825 . . . . . . . 8 𝑗(𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))))
131 nfra1 2936 . . . . . . . . . . 11 𝑡𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)
13219, 131nfan 1825 . . . . . . . . . 10 𝑡(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
133 nfcv 2761 . . . . . . . . . . 11 𝑡(0...𝑛)
134 nfra1 2936 . . . . . . . . . . . 12 𝑡𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1)
135 nfra1 2936 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛)
136 nfra1 2936 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)
137134, 135, 136nf3an 1828 . . . . . . . . . . 11 𝑡(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
138133, 137nfral 2940 . . . . . . . . . 10 𝑡𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
139132, 110, 138nf3an 1828 . . . . . . . . 9 𝑡((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))
14014, 139nfan 1825 . . . . . . . 8 𝑡(𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))))
141 eqid 2621 . . . . . . . 8 (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)}) = (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)})
142 uniexg 6908 . . . . . . . . . . 11 (𝐽 ∈ Comp → 𝐽 ∈ V)
14369, 142syl 17 . . . . . . . . . 10 (𝜑 𝐽 ∈ V)
14425, 143syl5eqel 2702 . . . . . . . . 9 (𝜑𝑇 ∈ V)
145144adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝑇 ∈ V)
14628adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝐹:𝑇⟶ℝ)
147 stoweidlem60.16 . . . . . . . . . 10 (𝜑 → ∀𝑡𝑇 0 ≤ (𝐹𝑡))
148147r19.21bi 2927 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
149148adantlr 750 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑡𝑇) → 0 ≤ (𝐹𝑡))
150 simpr1r 1117 . . . . . . . . 9 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
151150r19.21bi 2927 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑡𝑇) → (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
1523adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝐸 ∈ ℝ+)
15394adantr 481 . . . . . . . 8 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → 𝐸 < (1 / 3))
154 simpll 789 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝜑)
155 simplr2 1102 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝑥:(0...𝑛)⟶𝐴)
156 simpr 477 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → 𝑗 ∈ (0...𝑛))
157 simp1 1059 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → 𝜑)
158 ffvelrn 6313 . . . . . . . . . . 11 ((𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
1591583adant1 1077 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
16082sselda 3583 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗) ∈ 𝐶)
16124, 25, 26, 160fcnre 38664 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗):𝑇⟶ℝ)
162157, 159, 161syl2anc 692 . . . . . . . . 9 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗):𝑇⟶ℝ)
163154, 155, 156, 162syl3anc 1323 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛)) → (𝑥𝑗):𝑇⟶ℝ)
164 simp1r3 1157 . . . . . . . . . 10 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))
165 r19.26-3 3059 . . . . . . . . . . 11 (∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)) ↔ (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))
166165simp1bi 1074 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1))
167 simpl 473 . . . . . . . . . . . 12 ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥𝑗)‘𝑡))
168167ralimi 2947 . . . . . . . . . . 11 (∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
169168ralimi 2947 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
170164, 166, 1693syl 18 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
171 simp2 1060 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → 𝑗 ∈ (0...𝑛))
172 simp3 1061 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → 𝑡𝑇)
173 rspa 2925 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
174173r19.21bi 2927 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → 0 ≤ ((𝑥𝑗)‘𝑡))
175170, 171, 172, 174syl21anc 1322 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → 0 ≤ ((𝑥𝑗)‘𝑡))
176 simpr 477 . . . . . . . . . . . 12 ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ((𝑥𝑗)‘𝑡) ≤ 1)
177176ralimi 2947 . . . . . . . . . . 11 (∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
178177ralimi 2947 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
179164, 166, 1783syl 18 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
180 rspa 2925 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
181180r19.21bi 2927 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → ((𝑥𝑗)‘𝑡) ≤ 1)
182179, 171, 172, 181syl21anc 1322 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡𝑇) → ((𝑥𝑗)‘𝑡) ≤ 1)
183 simp1r3 1157 . . . . . . . . . 10 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷𝑗)) → ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))
184165simp2bi 1075 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
185183, 184syl 17 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷𝑗)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
186 simp2 1060 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷𝑗)) → 𝑗 ∈ (0...𝑛))
187 simp3 1061 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷𝑗)) → 𝑡 ∈ (𝐷𝑗))
188 rspa 2925 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
189188r19.21bi 2927 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐷𝑗)) → ((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
190185, 186, 187, 189syl21anc 1322 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐷𝑗)) → ((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
191 simp1r3 1157 . . . . . . . . . 10 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵𝑗)) → ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))
192165simp3bi 1076 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
193191, 192syl 17 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵𝑗)) → ∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
194 simp2 1060 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑗 ∈ (0...𝑛))
195 simp3 1061 . . . . . . . . 9 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵𝑗)) → 𝑡 ∈ (𝐵𝑗))
196 rspa 2925 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
197196r19.21bi 2927 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐵𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
198193, 194, 195, 197syl21anc 1322 . . . . . . . 8 (((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) ∧ 𝑗 ∈ (0...𝑛) ∧ 𝑡 ∈ (𝐵𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
19968, 130, 140, 77, 78, 141, 108, 145, 146, 149, 151, 152, 153, 163, 175, 182, 190, 198stoweidlem34 39555 . . . . . . 7 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → ∀𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡))))
200 nfmpt1 4707 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
201200nfeq2 2776 . . . . . . . . 9 𝑡 𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
202 fveq1 6147 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (𝑔𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡))
203202breq1d 4623 . . . . . . . . . . . 12 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ↔ ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸)))
204202breq2d 4625 . . . . . . . . . . . 12 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡) ↔ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡)))
205203, 204anbi12d 746 . . . . . . . . . . 11 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)) ↔ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡))))
206205anbi2d 739 . . . . . . . . . 10 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) ↔ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡)))))
207206rexbidv 3045 . . . . . . . . 9 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) ↔ ∃𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡)))))
208201, 207ralbid 2977 . . . . . . . 8 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (∀𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) ↔ ∀𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡)))))
209208rspcev 3295 . . . . . . 7 (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) ∈ 𝐴 ∧ ∀𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ (((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡)))) → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
210124, 199, 209syl2anc 692 . . . . . 6 ((𝜑 ∧ ((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)))) → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
211210ex 450 . . . . 5 (𝜑 → (((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))) → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)))))
2122112eximdv 1845 . . . 4 (𝜑 → (∃𝑛𝑥((𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)) ∧ 𝑥:(0...𝑛)⟶𝐴 ∧ ∀𝑗 ∈ (0...𝑛)(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))) → ∃𝑛𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)))))
213106, 212mpd 15 . . 3 (𝜑 → ∃𝑛𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
214 idd 24 . . . 4 (𝜑 → (∃𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) → ∃𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)))))
215214exlimdv 1858 . . 3 (𝜑 → (∃𝑛𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) → ∃𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)))))
216213, 215mpd 15 . 2 (𝜑 → ∃𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
217 idd 24 . . 3 (𝜑 → (∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)))))
218217exlimdv 1858 . 2 (𝜑 → (∃𝑥𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))) → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡)))))
219216, 218mpd 15 1 (𝜑 → ∃𝑔𝐴𝑡𝑇𝑗 ∈ ℝ ((((𝑗 − (4 / 3)) · 𝐸) < (𝐹𝑡) ∧ (𝐹𝑡) ≤ ((𝑗 − (1 / 3)) · 𝐸)) ∧ ((𝑔𝑡) < ((𝑗 + (1 / 3)) · 𝐸) ∧ ((𝑗 − (4 / 3)) · 𝐸) < (𝑔𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wnf 1705  wcel 1987  wnfc 2748  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  wss 3555  c0 3891   cuni 4402   class class class wbr 4613  cmpt 4673  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  cr 9879  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  cn 10964  3c3 11015  4c4 11016  +crp 11776  (,)cioo 12117  ...cfz 12268  Σcsu 14350  topGenctg 16019   Cn ccn 20938  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-cn 20941  df-cnp 20942  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-xms 22035  df-ms 22036  df-tms 22037
This theorem is referenced by:  stoweidlem61  39582
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