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Theorem stoweidlem60 42222
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 11633 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
21adantl 482 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝑚 ∈ ℝ)
3 stoweidlem60.17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ ℝ+)
43rpred 12419 . . . . . . . . . . . . 13 (𝜑𝐸 ∈ ℝ)
54adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ∈ ℝ)
63rpne0d 12424 . . . . . . . . . . . . 13 (𝜑𝐸 ≠ 0)
76adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐸 ≠ 0)
82, 5, 7redivcld 11456 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝑚 / 𝐸) ∈ ℝ)
9 1red 10630 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 1 ∈ ℝ)
108, 9readdcld 10658 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
1110adantr 481 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) → ((𝑚 / 𝐸) + 1) ∈ ℝ)
12 arch 11882 . . . . . . . . 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 1906 . . . . . . . . . . . . . . 15 𝑡 𝑚 ∈ ℕ
1614, 15nfan 1891 . . . . . . . . . . . . . 14 𝑡(𝜑𝑚 ∈ ℕ)
17 nfra1 3216 . . . . . . . . . . . . . 14 𝑡𝑡𝑇 (𝐹𝑡) < 𝑚
1816, 17nfan 1891 . . . . . . . . . . . . 13 𝑡((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
19 nfv 1906 . . . . . . . . . . . . 13 𝑡 𝑛 ∈ ℕ
2018, 19nfan 1891 . . . . . . . . . . . 12 𝑡(((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ)
21 nfv 1906 . . . . . . . . . . . 12 𝑡((𝑚 / 𝐸) + 1) < 𝑛
2220, 21nfan 1891 . . . . . . . . . . 11 𝑡((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛)
23 simp-5l 781 . . . . . . . . . . . . . 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 41159 . . . . . . . . . . . . . . 15 (𝜑𝐹:𝑇⟶ℝ)
2928ffvelrnda 6843 . . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . 14 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℕ)
3231nnred 11641 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 ∈ ℝ)
33 simpllr 772 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℕ)
3433nnred 11641 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑛 ∈ ℝ)
35 1red 10630 . . . . . . . . . . . . . . 15 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 1 ∈ ℝ)
3634, 35resubcld 11056 . . . . . . . . . . . . . 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 10659 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → ((𝑛 − 1) · 𝐸) ∈ ℝ)
39 simpllr 772 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
4039r19.21bi 3205 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → (𝐹𝑡) < 𝑚)
41 simplr 765 . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝜑)
44 simpl2 1184 . . . . . . . . . . . . . . . . . 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 10630 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 1 ∈ ℝ)
47 simpl3 1185 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℕ)
4847nnred 11641 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑛 ∈ ℝ)
4945, 46, 48ltaddsubd 11228 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (((𝑚 / 𝐸) + 1) < 𝑛 ↔ (𝑚 / 𝐸) < (𝑛 − 1)))
5042, 49mpbid 233 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑚 / 𝐸) < (𝑛 − 1))
5113ad2ant2 1126 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℝ)
5251adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 ∈ ℝ)
5348, 46resubcld 11056 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → (𝑛 − 1) ∈ ℝ)
5443ad2ant1 1125 . . . . . . . . . . . . . . . . 17 ((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) → 𝐸 ∈ ℝ)
5554adantr 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝐸 ∈ ℝ)
563rpgt0d 12422 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝐸)
5743, 56syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 0 < 𝐸)
58 ltdivmul2 11505 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ ℝ ∧ (𝑛 − 1) ∈ ℝ ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
5952, 53, 55, 57, 58syl112anc 1366 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → ((𝑚 / 𝐸) < (𝑛 − 1) ↔ 𝑚 < ((𝑛 − 1) · 𝐸)))
6050, 59mpbid 233 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) → 𝑚 < ((𝑛 − 1) · 𝐸))
6123, 31, 33, 41, 60syl31anc 1365 . . . . . . . . . . . . 13 ((((((𝜑𝑚 ∈ ℕ) ∧ ∀𝑡𝑇 (𝐹𝑡) < 𝑚) ∧ 𝑛 ∈ ℕ) ∧ ((𝑚 / 𝐸) + 1) < 𝑛) ∧ 𝑡𝑇) → 𝑚 < ((𝑛 − 1) · 𝐸))
6230, 32, 38, 40, 61lttrd 10789 . . . . . . . . . . . 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 3213 . . . . . . . . . 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 3271 . . . . . . . 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 41170 . . . . . . 7 (𝜑 → ∃𝑚 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < 𝑚)
7267, 71r19.29a 3286 . . . . . 6 (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
73 df-rex 3141 . . . . . 6 (∃𝑛 ∈ ℕ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)))
7472, 73sylib 219 . . . . 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 1891 . . . . . . . . . . 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 2818 . . . . . . . . . . 11 {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)} = {𝑦𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑦𝑡) ∧ (𝑦𝑡) ≤ 1)}
80 eqid 2818 . . . . . . . . . . 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 1169 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
86 stoweidlem60.12 . . . . . . . . . . . 12 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
87863adant1r 1169 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
88 stoweidlem60.13 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
8988adantlr 711 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
90 stoweidlem60.14 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑡𝑇𝑟𝑡)) → ∃𝑞𝐴 (𝑞𝑟) ≠ (𝑞𝑡))
9190adantlr 711 . . . . . . . . . . 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 42221 . . . . . . . . . 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 713 . . . . . . . . 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 1945 . . . . . . . . 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 1087 . . . . . . . . 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 1839 . . . . . . . 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 235 . . . . . . 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 1909 . . . . 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 1222 . . . . . . . 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 1187 . . . . . . . 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 1906 . . . . . . . . . 10 𝑡 𝑥:(0...𝑛)⟶𝐴
11114, 19, 110nf3an 1893 . . . . . . . . 9 𝑡(𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴)
112 simp2 1129 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑛 ∈ ℕ)
113 simp3 1130 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝑥:(0...𝑛)⟶𝐴)
114 simp1 1128 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝜑)
115114, 84syl3an1 1155 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) + (𝑔𝑡))) ∈ 𝐴)
116114, 86syl3an1 1155 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
117883ad2antl1 1177 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑦 ∈ ℝ) → (𝑡𝑇𝑦) ∈ 𝐴)
11833ad2ant1 1125 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ+)
119118rpred 12419 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → 𝐸 ∈ ℝ)
12082sselda 3964 . . . . . . . . . . 11 ((𝜑𝑓𝐴) → 𝑓𝐶)
12124, 25, 26, 120fcnre 41159 . . . . . . . . . 10 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
1221213ad2antl1 1177 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
123111, 112, 113, 115, 116, 117, 119, 122stoweidlem17 42179 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑥:(0...𝑛)⟶𝐴) → (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) ∈ 𝐴)
124107, 108, 109, 123syl3anc 1363 . . . . . . 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 1906 . . . . . . . . 9 𝑗𝜑
126 nfv 1906 . . . . . . . . . 10 𝑗(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
127 nfv 1906 . . . . . . . . . 10 𝑗 𝑥:(0...𝑛)⟶𝐴
128 nfra1 3216 . . . . . . . . . 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 1893 . . . . . . . . 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 1891 . . . . . . . 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 3216 . . . . . . . . . . 11 𝑡𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸)
13219, 131nfan 1891 . . . . . . . . . 10 𝑡(𝑛 ∈ ℕ ∧ ∀𝑡𝑇 (𝐹𝑡) < ((𝑛 − 1) · 𝐸))
133 nfcv 2974 . . . . . . . . . . 11 𝑡(0...𝑛)
134 nfra1 3216 . . . . . . . . . . . 12 𝑡𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1)
135 nfra1 3216 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛)
136 nfra1 3216 . . . . . . . . . . . 12 𝑡𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡)
137134, 135, 136nf3an 1893 . . . . . . . . . . 11 𝑡(∀𝑡𝑇 (0 ≤ ((𝑥𝑗)‘𝑡) ∧ ((𝑥𝑗)‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
138133, 137nfralw 3222 . . . . . . . . . 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 1893 . . . . . . . . 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 1891 . . . . . . . 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 2818 . . . . . . . 8 (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)}) = (𝑡𝑇 ↦ {𝑗 ∈ (1...𝑛) ∣ 𝑡 ∈ (𝐷𝑗)})
14269uniexd 7457 . . . . . . . . . 10 (𝜑 𝐽 ∈ V)
14325, 142eqeltrid 2914 . . . . . . . . 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 3205 . . . . . . . . 9 ((𝜑𝑡𝑇) → 0 ≤ (𝐹𝑡))
148147adantlr 711 . . . . . . . 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 1223 . . . . . . . . 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 3205 . . . . . . . 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 763 . . . . . . . . 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 1208 . . . . . . . . 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 1128 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → 𝜑)
157 ffvelrn 6841 . . . . . . . . . . 11 ((𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
1581573adant1 1122 . . . . . . . . . 10 ((𝜑𝑥:(0...𝑛)⟶𝐴𝑗 ∈ (0...𝑛)) → (𝑥𝑗) ∈ 𝐴)
15982sselda 3964 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑗) ∈ 𝐴) → (𝑥𝑗) ∈ 𝐶)
16024, 25, 26, 159fcnre 41159 . . . . . . . . . 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 1363 . . . . . . . 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 1263 . . . . . . . . . 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 3169 . . . . . . . . . . 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 1137 . . . . . . . . . 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 3158 . . . . . . . . . 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 1129 . . . . . . . . 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 1130 . . . . . . . . 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 3203 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡))
172171r19.21bi 3205 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 0 ≤ ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → 0 ≤ ((𝑥𝑗)‘𝑡))
173168, 169, 170, 172syl21anc 833 . . . . . . . 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 3158 . . . . . . . . . 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 3203 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1)
178177r19.21bi 3205 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡𝑇 ((𝑥𝑗)‘𝑡) ≤ 1 ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡𝑇) → ((𝑥𝑗)‘𝑡) ≤ 1)
179176, 169, 170, 178syl21anc 833 . . . . . . . 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 1263 . . . . . . . . . 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 1138 . . . . . . . . . 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 1129 . . . . . . . . 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 1130 . . . . . . . . 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 3203 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
186185r19.21bi 3205 . . . . . . . . 9 (((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐷𝑗)((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛) ∧ 𝑗 ∈ (0...𝑛)) ∧ 𝑡 ∈ (𝐷𝑗)) → ((𝑥𝑗)‘𝑡) < (𝐸 / 𝑛))
187182, 183, 184, 186syl21anc 833 . . . . . . . 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 1263 . . . . . . . . . 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 1139 . . . . . . . . . 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 1129 . . . . . . . . 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 1130 . . . . . . . . 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 3203 . . . . . . . . . 10 ((∀𝑗 ∈ (0...𝑛)∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡) ∧ 𝑗 ∈ (0...𝑛)) → ∀𝑡 ∈ (𝐵𝑗)(1 − (𝐸 / 𝑛)) < ((𝑥𝑗)‘𝑡))
194193r19.21bi 3205 . . . . . . . . 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 833 . . . . . . . 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 42196 . . . . . . 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 5155 . . . . . . . . . 10 𝑡(𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
198197nfeq2 2992 . . . . . . . . 9 𝑡 𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))
199 fveq1 6662 . . . . . . . . . . . . 13 (𝑔 = (𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡))) → (𝑔𝑡) = ((𝑡𝑇 ↦ Σ𝑖 ∈ (0...𝑛)(𝐸 · ((𝑥𝑖)‘𝑡)))‘𝑡))
200199breq1d 5067 . . . . . . . . . . . 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 5069 . . . . . . . . . . . 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 630 . . . . . . . . . . 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 628 . . . . . . . . . 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 3294 . . . . . . . . 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 3228 . . . . . . . 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 3620 . . . . . . 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 1911 . . . 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 1925 . . 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 1925 . 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 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wnf 1775  wcel 2105  wnfc 2958  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  c0 4288   cuni 4830   class class class wbr 5057  cmpt 5137  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  3c3 11681  4c4 11682  +crp 12377  (,)cioo 12726  ...cfz 12880  Σcsu 15030  topGenctg 16699   Cn ccn 21760  Compccmp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-cn 21763  df-cnp 21764  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-xms 22857  df-ms 22858  df-tms 22859
This theorem is referenced by:  stoweidlem61  42223
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