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Theorem stoweidlem60 40595
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 11065 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
21adantl 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3 stoweidlem60.17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ ℝ+)
43rpred 11910 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ℝ)
54adantr 480 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
63rpne0d 11915 . . . . . . . . . . . . 13 (𝜑𝐸 ≠ 0)
76adantr 480 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ≠ 0)
82, 5, 7redivcld 10891 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9 1red 10093 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℝ)
108, 9readdcld 10107 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
1110adantr 480 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12 arch 11327 . . . . . . . . 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 1883 . . . . . . . . . . . . . . 15 𝑡 𝑚 ∈ ℕ
1614, 15nfan 1868 . . . . . . . . . . . . . 14 𝑡(𝜑𝑚 ∈ ℕ)
17 nfra1 2970 . . . . . . . . . . . . . 14 𝑡𝑡𝑇 (𝐹𝑡) < 𝑚
1816, 17nfan 1868 . . . . . . . . . . . . 13 𝑡((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
19 nfv 1883 . . . . . . . . . . . . 13 𝑡 𝑛 ∈ ℕ
2018, 19nfan 1868 . . . . . . . . . . . 12 𝑡(((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21 nfv 1883 . . . . . . . . . . . 12 𝑡((𝑚 / 𝐸) + 1) < 𝑛
2220, 21nfan 1868 . . . . . . . . . . 11 𝑡((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23 simp-5l 825 . . . . . . . . . . . . . 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 39498 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑇⟶ℝ)
2928ffvelrnda 6399 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
3023, 29sylancom 702 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
31 simp-5r 826 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ)
3231nnred 11073 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℝ)
33 simpllr 815 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℕ)
3433nnred 11073 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℝ)
35 1red 10093 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 1 ∈ ℝ)
3634, 35resubcld 10496 . . . . . . . . . . . . . 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 10108 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39 simpllr 815 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
4039r19.21bi 2961 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < 𝑚)
41 simplr 807 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42 simpr 476 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43 simpl1 1084 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44 simpl2 1085 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
4543, 44, 8syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46 1red 10093 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47 simpl3 1086 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
4847nnred 11073 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
4945, 46, 48ltaddsubd 10665 . . . . . . . . . . . . . . . 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 1103 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
5251adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
5348, 46resubcld 10496 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
5443ad2ant1 1102 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
5554adantr 480 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
563rpgt0d 11913 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝐸)
5743, 56syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58 ltdivmul2 10938 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℝ ∧ (𝑛 − 1) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
5952, 53, 55, 57, 58syl112anc 1370 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 < ((𝑛 − 1) · 𝐸))
6230, 32, 38, 40, 61lttrd 10236 . . . . . . . . . . . 12 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
6362ex 449 . . . . . . . . . . 11 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑡𝑇 → (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6422, 63ralrimi 2986 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
6564ex 449 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) → (((𝑚 / 𝐸) + 1) < 𝑛 → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6665reximdva 3046 . . . . . . . 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 39509 . . . . . . 7 (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
7267, 71r19.29a 3107 . . . . . 6 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
73 df-rex 2947 . . . . . 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 476 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
7614, 19nfan 1868 . . . . . . . . . . 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 2651 . . . . . . . . . . 11 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
80 eqid 2651 . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐽 ∈ Comp)
82 stoweidlem60.10 . . . . . . . . . . . 12 (𝜑𝐴𝐶)
8382adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐴𝐶)
84 stoweidlem60.11 . . . . . . . . . . . 12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
85843adant1r 1359 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
86 stoweidlem60.12 . . . . . . . . . . . 12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
87863adant1r 1359 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
88 stoweidlem60.13 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
8988adantlr 751 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
90 stoweidlem60.14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9190adantlr 751 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9227adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐹𝐶)
933adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐸 ∈ ℝ+)
94 stoweidlem60.18 . . . . . . . . . . . 12 (𝜑𝐸 < (1 / 3))
9594adantr 480 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝐸 < (1 / 3))
96 simpr 476 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9768, 76, 24, 25, 26, 77, 78, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 96stoweidlem59 40594 . . . . . . . . . 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 753 . . . . . . . . 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 1921 . . . . . . . . 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 699 . . . . . . . 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 1059 . . . . . . . . 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 1814 . . . . . . . 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 449 . . . . . 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 1886 . . . . 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 472 . . . . . . . 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 1138 . . . . . . . 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 1088 . . . . . . . 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 1883 . . . . . . . . . 10 𝑡 𝑥:(0...𝑛)⟶𝐴
11114, 19, 110nf3an 1871 . . . . . . . . 9 𝑡(𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112 simp2 1082 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113 simp3 1083 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114 simp1 1081 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115114, 84syl3an1 1399 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116114, 86syl3an1 1399 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
117883ad2antl1 1243 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
11833ad2ant1 1102 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ+)
119118rpred 11910 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
12082sselda 3636 . . . . . . . . . . 11 ((𝜑𝑓𝐴) → 𝑓𝐶)
12124, 25, 26, 120fcnre 39498 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1221213ad2antl1 1243 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
123111, 112, 113, 115, 116, 117, 119, 122stoweidlem17 40552 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) ∈ 𝐴)
124107, 108, 109, 123syl3anc 1366 . . . . . . 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 1883 . . . . . . . . 9 𝑗𝜑
126 nfv 1883 . . . . . . . . . 10 𝑗(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
127 nfv 1883 . . . . . . . . . 10 𝑗 𝑥:(0...𝑛)⟶𝐴
128 nfra1 2970 . . . . . . . . . 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 1871 . . . . . . . . 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 1868 . . . . . . . 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 2970 . . . . . . . . . . 11 𝑡𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)
13219, 131nfan 1868 . . . . . . . . . 10 𝑡(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
133 nfcv 2793 . . . . . . . . . . 11 𝑡(0...𝑛)
134 nfra1 2970 . . . . . . . . . . . 12 𝑡𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1)
135 nfra1 2970 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛)
136 nfra1 2970 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)
137134, 135, 136nf3an 1871 . . . . . . . . . . 11 𝑡(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
138133, 137nfral 2974 . . . . . . . . . 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 1871 . . . . . . . . 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 1868 . . . . . . . 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 2651 . . . . . . . 8 (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)}) = (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)})
142 uniexg 6997 . . . . . . . . . . 11 (𝐽 ∈ Comp → 𝐽 ∈ V)
14369, 142syl 17 . . . . . . . . . 10 (𝜑 𝐽 ∈ V)
14425, 143syl5eqel 2734 . . . . . . . . 9 (𝜑𝑇 ∈ V)
145144adantr 480 . . . . . . . 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 480 . . . . . . . 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 2961 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
149148adantlr 751 . . . . . . . 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 1139 . . . . . . . . 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 2961 . . . . . . . 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 480 . . . . . . . 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 480 . . . . . . . 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 805 . . . . . . . . 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 1124 . . . . . . . . 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 476 . . . . . . . . 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 1081 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → 𝜑)
158 ffvelrn 6397 . . . . . . . . . . 11 ((𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
1591583adant1 1099 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
16082sselda 3636 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗) ∈ 𝐶)
16124, 25, 26, 160fcnre 39498 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗):𝑇⟶ℝ)
162157, 159, 161syl2anc 694 . . . . . . . . 9 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗):𝑇⟶ℝ)
163154, 155, 156, 162syl3anc 1366 . . . . . . . 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 1179 . . . . . . . . . 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 3095 . . . . . . . . . . 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 1096 . . . . . . . . . 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 472 . . . . . . . . . . 11 ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥𝑗)‘𝑡))
1681672ralimi 2982 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
169164, 166, 1683syl 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))
170 simp2 1082 . . . . . . . . 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))
171 simp3 1083 . . . . . . . . 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)
172 rspa 2959 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
173172r19.21bi 2961 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → 0 ≤ ((𝑥𝑗)‘𝑡))
174169, 170, 171, 173syl21anc 1365 . . . . . . . 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))
175 simpr 476 . . . . . . . . . . 11 ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ((𝑥𝑗)‘𝑡) ≤ 1)
1761752ralimi 2982 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
177164, 166, 1763syl 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)
178 rspa 2959 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
179178r19.21bi 2961 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → ((𝑥𝑗)‘𝑡) ≤ 1)
180177, 170, 171, 179syl21anc 1365 . . . . . . . 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)
181 simp1r3 1179 . . . . . . . . . 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)))
182165simp2bi 1097 . . . . . . . . . 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))
183181, 182syl 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))
184 simp2 1082 . . . . . . . . 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))
185 simp3 1083 . . . . . . . . 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))
186 rspa 2959 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
187186r19.21bi 2961 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐷𝑗)) → ((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
188183, 184, 185, 187syl21anc 1365 . . . . . . . 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))
189 simp1r3 1179 . . . . . . . . . 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)))
190165simp3bi 1098 . . . . . . . . . 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))
191189, 190syl 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))
192 simp2 1082 . . . . . . . . 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))
193 simp3 1083 . . . . . . . . 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))
194 rspa 2959 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
195194r19.21bi 2961 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐵𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
196191, 192, 193, 195syl21anc 1365 . . . . . . . 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))
19768, 130, 140, 77, 78, 141, 108, 145, 146, 149, 151, 152, 153, 163, 174, 180, 188, 196stoweidlem34 40569 . . . . . . 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))))
198 nfmpt1 4780 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
199198nfeq2 2809 . . . . . . . . 9 𝑡 𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
200 fveq1 6228 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (𝑔𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡))
201200breq1d 4695 . . . . . . . . . . . 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)))
202200breq2d 4697 . . . . . . . . . . . 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)))
203201, 202anbi12d 747 . . . . . . . . . . 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))))
204203anbi2d 740 . . . . . . . . . 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)))))
205204rexbidv 3081 . . . . . . . . 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)))))
206199, 205ralbid 3012 . . . . . . . 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)))))
207206rspcev 3340 . . . . . . 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))))
208124, 197, 207syl2anc 694 . . . . . 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))))
209208ex 449 . . . . 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)))))
2102092eximdv 1888 . . . 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)))))
211106, 210mpd 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))))
212 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)))))
213212exlimdv 1901 . . 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)))))
214211, 213mpd 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))))
215 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)))))
216215exlimdv 1901 . 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)))))
217214, 216mpd 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 383  w3a 1054   = wceq 1523  wex 1744  wnf 1748  wcel 2030  wnfc 2780  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948   cuni 4468   class class class wbr 4685  cmpt 4762  ran crn 5144  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  3c3 11109  4c4 11110  +crp 11870  (,)cioo 12213  ...cfz 12364  Σcsu 14460  topGenctg 16145   Cn ccn 21076  Compccmp 21237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-rlim 14264  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-cn 21079  df-cnp 21080  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-xms 22172  df-ms 22173  df-tms 22174
This theorem is referenced by:  stoweidlem61  40596
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