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Theorem stoweidlem60 44291
Description: This lemma proves that there exists a function g as in the proof in [BrosowskiDeutsh] p. 91 (this parte of the proof actually spans through pages 91-92): g is in the subalgebra, and for all 𝑡 in 𝑇, there is a 𝑗 such that (j-4/3)*ε < f(t) <= (j-1/3)*ε and (j-4/3)*ε < g(t) < (j+1/3)*ε. Here 𝐹 is used to represent f in the paper, and 𝐸 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem60.1 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 12160 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
21adantl 482 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3 stoweidlem60.17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ ℝ+)
43rpred 12957 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ℝ)
54adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
63rpne0d 12962 . . . . . . . . . . . . 13 (𝜑𝐸 ≠ 0)
76adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ≠ 0)
82, 5, 7redivcld 11983 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9 1red 11156 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℝ)
108, 9readdcld 11184 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
1110adantr 481 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12 arch 12410 . . . . . . . . 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 1917 . . . . . . . . . . . . . . 15 𝑡 𝑚 ∈ ℕ
1614, 15nfan 1902 . . . . . . . . . . . . . 14 𝑡(𝜑𝑚 ∈ ℕ)
17 nfra1 3267 . . . . . . . . . . . . . 14 𝑡𝑡𝑇 (𝐹𝑡) < 𝑚
1816, 17nfan 1902 . . . . . . . . . . . . 13 𝑡((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
19 nfv 1917 . . . . . . . . . . . . 13 𝑡 𝑛 ∈ ℕ
2018, 19nfan 1902 . . . . . . . . . . . 12 𝑡(((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21 nfv 1917 . . . . . . . . . . . 12 𝑡((𝑚 / 𝐸) + 1) < 𝑛
2220, 21nfan 1902 . . . . . . . . . . 11 𝑡((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23 simp-5l 783 . . . . . . . . . . . . . 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 43220 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑇⟶ℝ)
2928ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((𝜑𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
3023, 29sylancom 588 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) ∈ ℝ)
31 simp-5r 784 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ)
3231nnred 12168 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℝ)
33 simpllr 774 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℕ)
3433nnred 12168 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℝ)
35 1red 11156 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 1 ∈ ℝ)
3634, 35resubcld 11583 . . . . . . . . . . . . . 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 11185 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39 simpllr 774 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
4039r19.21bi 3234 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < 𝑚)
41 simplr 767 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑚 / 𝐸) + 1) < 𝑛)
42 simpr 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) + 1) < 𝑛)
43 simpl1 1191 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44 simpl2 1192 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℕ)
4543, 44, 8syl2anc 584 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) ∈ ℝ)
46 1red 11156 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47 simpl3 1193 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
4847nnred 12168 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
4945, 46, 48ltaddsubd 11755 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (((𝑚 / 𝐸) + 1) < 𝑛 ↔ (𝑚 / 𝐸) < (𝑛 − 1)))
5042, 49mpbid 231 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) < (𝑛 − 1))
5113ad2ant2 1134 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
5251adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
5348, 46resubcld 11583 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
5443ad2ant1 1133 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
5554adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
563rpgt0d 12960 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝐸)
5743, 56syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58 ltdivmul2 12032 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℝ ∧ (𝑛 − 1) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
5952, 53, 55, 57, 58syl112anc 1374 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
6050, 59mpbid 231 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 < ((𝑛 − 1) · 𝐸))
6123, 31, 33, 41, 60syl31anc 1373 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 < ((𝑛 − 1) · 𝐸))
6230, 32, 38, 40, 61lttrd 11316 . . . . . . . . . . . 12 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
6362ex 413 . . . . . . . . . . 11 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑡𝑇 → (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6422, 63ralrimi 3240 . . . . . . . . . 10 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
6564ex 413 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) → (((𝑚 / 𝐸) + 1) < 𝑛 → ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
6665reximdva 3165 . . . . . . . 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 43231 . . . . . . 7 (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
7267, 71r19.29a 3159 . . . . . 6 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
73 df-rex 3074 . . . . . 6 (∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
7472, 73sylib 217 . . . . 5 (𝜑 → ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
75 simpr 485 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))) → (𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
7614, 19nfan 1902 . . . . . . . . . . 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 2736 . . . . . . . . . . 11 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
80 eqid 2736 . . . . . . . . . . 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 1177 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
86 stoweidlem60.12 . . . . . . . . . . . 12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
87863adant1r 1177 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
88 stoweidlem60.13 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
8988adantlr 713 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
90 stoweidlem60.14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9190adantlr 713 . . . . . . . . . . 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 485 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
9768, 76, 24, 25, 26, 77, 78, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 96stoweidlem59 44290 . . . . . . . . . 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 715 . . . . . . . . 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 1957 . . . . . . . . 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 583 . . . . . . . 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 1095 . . . . . . . . 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 1850 . . . . . . . 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 233 . . . . . . 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 413 . . . . . 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 1920 . . . . 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 483 . . . . . . . 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 1230 . . . . . . . 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 1195 . . . . . . . 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 1917 . . . . . . . . . 10 𝑡 𝑥:(0...𝑛)⟶𝐴
11114, 19, 110nf3an 1904 . . . . . . . . 9 𝑡(𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112 simp2 1137 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113 simp3 1138 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114 simp1 1136 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115114, 84syl3an1 1163 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116114, 86syl3an1 1163 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
117883ad2antl1 1185 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
11833ad2ant1 1133 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ+)
119118rpred 12957 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
12082sselda 3944 . . . . . . . . . . 11 ((𝜑𝑓𝐴) → 𝑓𝐶)
12124, 25, 26, 120fcnre 43220 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1221213ad2antl1 1185 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
123111, 112, 113, 115, 116, 117, 119, 122stoweidlem17 44248 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) ∈ 𝐴)
124107, 108, 109, 123syl3anc 1371 . . . . . . 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 1917 . . . . . . . . 9 𝑗𝜑
126 nfv 1917 . . . . . . . . . 10 𝑗(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
127 nfv 1917 . . . . . . . . . 10 𝑗 𝑥:(0...𝑛)⟶𝐴
128 nfra1 3267 . . . . . . . . . 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 1904 . . . . . . . . 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 1902 . . . . . . . 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 3267 . . . . . . . . . . 11 𝑡𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)
13219, 131nfan 1902 . . . . . . . . . 10 𝑡(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
133 nfcv 2907 . . . . . . . . . . 11 𝑡(0...𝑛)
134 nfra1 3267 . . . . . . . . . . . 12 𝑡𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1)
135 nfra1 3267 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛)
136 nfra1 3267 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)
137134, 135, 136nf3an 1904 . . . . . . . . . . 11 𝑡(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
138133, 137nfralw 3294 . . . . . . . . . 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 1904 . . . . . . . . 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 1902 . . . . . . . 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 2736 . . . . . . . 8 (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)}) = (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)})
14269uniexd 7679 . . . . . . . . . 10 (𝜑 𝐽 ∈ V)
14325, 142eqeltrid 2842 . . . . . . . . 9 (𝜑𝑇 ∈ V)
144143adantr 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)
14528adantr 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)
146 stoweidlem60.16 . . . . . . . . . 10 (𝜑 → ∀𝑡𝑇 0 ≤ (𝐹𝑡))
147146r19.21bi 3234 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
148147adantlr 713 . . . . . . . 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))
149 simpr1r 1231 . . . . . . . . 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))
150149r19.21bi 3234 . . . . . . . 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))
1513adantr 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+)
15294adantr 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))
153 simpll 765 . . . . . . . . 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)
154 simplr2 1216 . . . . . . . . 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)
155 simpr 485 . . . . . . . . 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))
156 simp1 1136 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → 𝜑)
157 ffvelcdm 7032 . . . . . . . . . . 11 ((𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
1581573adant1 1130 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
15982sselda 3944 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗) ∈ 𝐶)
16024, 25, 26, 159fcnre 43220 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗):𝑇⟶ℝ)
161156, 158, 160syl2anc 584 . . . . . . . . 9 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗):𝑇⟶ℝ)
162153, 154, 155, 161syl3anc 1371 . . . . . . . 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)
163 simp1r3 1271 . . . . . . . . . 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)))
164 r19.26-3 3115 . . . . . . . . . . 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)))
165164simp1bi 1145 . . . . . . . . . 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))
166 simpl 483 . . . . . . . . . . 11 ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → 0 ≤ ((𝑥𝑗)‘𝑡))
1671662ralimi 3126 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
168163, 165, 1673syl 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))
169 simp2 1137 . . . . . . . . 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))
170 simp3 1138 . . . . . . . . 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)
171 rspa 3231 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
172171r19.21bi 3234 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → 0 ≤ ((𝑥𝑗)‘𝑡))
173168, 169, 170, 172syl21anc 836 . . . . . . . 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))
174 simpr 485 . . . . . . . . . . 11 ((0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ((𝑥𝑗)‘𝑡) ≤ 1)
1751742ralimi 3126 . . . . . . . . . 10 (∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) → ∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
176163, 165, 1753syl 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)
177 rspa 3231 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
178177r19.21bi 3234 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → ((𝑥𝑗)‘𝑡) ≤ 1)
179176, 169, 170, 178syl21anc 836 . . . . . . . 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)
180 simp1r3 1271 . . . . . . . . . 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)))
181164simp2bi 1146 . . . . . . . . . 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))
182180, 181syl 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))
183 simp2 1137 . . . . . . . . 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))
184 simp3 1138 . . . . . . . . 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))
185 rspa 3231 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
186185r19.21bi 3234 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐷𝑗)) → ((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
187182, 183, 184, 186syl21anc 836 . . . . . . . 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))
188 simp1r3 1271 . . . . . . . . . 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)))
189164simp3bi 1147 . . . . . . . . . 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))
190188, 189syl 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))
191 simp2 1137 . . . . . . . . 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))
192 simp3 1138 . . . . . . . . 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))
193 rspa 3231 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
194193r19.21bi 3234 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐵𝑗)) → (1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
195190, 191, 192, 194syl21anc 836 . . . . . . . 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))
19668, 130, 140, 77, 78, 141, 108, 144, 145, 148, 150, 151, 152, 162, 173, 179, 187, 195stoweidlem34 44265 . . . . . . 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))))
197 nfmpt1 5213 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
198197nfeq2 2924 . . . . . . . . 9 𝑡 𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
199 fveq1 6841 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (𝑔𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡))
200199breq1d 5115 . . . . . . . . . . . 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)))
201199breq2d 5117 . . . . . . . . . . . 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)))
202200, 201anbi12d 631 . . . . . . . . . . 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))))
203202anbi2d 629 . . . . . . . . . 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)))))
204203rexbidv 3175 . . . . . . . . 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)))))
205198, 204ralbid 3256 . . . . . . . 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)))))
206205rspcev 3581 . . . . . . 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))))
207124, 196, 206syl2anc 584 . . . . . 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))))
208207ex 413 . . . . 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)))))
2092082eximdv 1922 . . . 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)))))
210106, 209mpd 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))))
211 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)))))
212211exlimdv 1936 . . 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)))))
213210, 212mpd 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))))
214 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)))))
215214exlimdv 1936 . 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)))))
216213, 215mpd 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wnf 1785  wcel 2106  wnfc 2887  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282   cuni 4865   class class class wbr 5105  cmpt 5188  ran crn 5634  wf 6492  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385   / cdiv 11812  cn 12153  3c3 12209  4c4 12210  +crp 12915  (,)cioo 13264  ...cfz 13424  Σcsu 15570  topGenctg 17319   Cn ccn 22575  Compccmp 22737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ioo 13268  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-rlim 15371  df-sum 15571  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-cn 22578  df-cnp 22579  df-cmp 22738  df-tx 22913  df-hmeo 23106  df-xms 23673  df-ms 23674  df-tms 23675
This theorem is referenced by:  stoweidlem61  44292
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