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Theorem stoweidlem54 38747
Description: There exists a function 𝑥 as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here 𝐷 is used to represent 𝐴 in the paper, because here 𝐴 is used for the subalgebra of functions. 𝐸 is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
stoweidlem54.1 𝑖𝜑
stoweidlem54.2 𝑡𝜑
stoweidlem54.3 𝑦𝜑
stoweidlem54.4 𝑤𝜑
stoweidlem54.5 𝑇 = 𝐽
stoweidlem54.6 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
stoweidlem54.7 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
stoweidlem54.8 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑦𝑖)‘𝑡)))
stoweidlem54.9 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
stoweidlem54.10 𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
stoweidlem54.11 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem54.12 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem54.13 (𝜑𝑀 ∈ ℕ)
stoweidlem54.14 (𝜑𝑊:(1...𝑀)⟶𝑉)
stoweidlem54.15 (𝜑𝐵𝑇)
stoweidlem54.16 (𝜑𝐷 ran 𝑊)
stoweidlem54.17 (𝜑𝐷𝑇)
stoweidlem54.18 (𝜑 → ∃𝑦(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))))
stoweidlem54.19 (𝜑𝑇 ∈ V)
stoweidlem54.20 (𝜑𝐸 ∈ ℝ+)
stoweidlem54.21 (𝜑𝐸 < (1 / 3))
Assertion
Ref Expression
stoweidlem54 (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
Distinct variable groups:   𝑓,𝑔,,𝑖,𝑡,𝑦,𝑇   𝐴,𝑓,𝑔,,𝑡,𝑦   𝐵,𝑓,𝑔,𝑖,𝑦   𝑓,𝐸,𝑔,𝑖,𝑦   𝑓,𝐹,𝑔   𝑓,𝑀,𝑔,,𝑖,𝑡   𝑓,𝑊,𝑔,𝑖   𝑓,𝑌,𝑔,𝑖   𝜑,𝑓,𝑔   𝑤,𝑖,𝑡,𝑦,𝑇   𝐷,𝑖,𝑦   𝑥,𝑡,𝑦,𝐴   𝑤,𝐵   𝑤,𝐸   𝑤,𝑀   𝑤,𝑊   𝑤,𝑌   𝑥,𝐵   𝑥,𝐷   𝑥,𝐸   𝑥,𝑀   𝑥,𝑃   𝑥,𝑇
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤,𝑡,𝑒,,𝑖)   𝐴(𝑤,𝑒,𝑖)   𝐵(𝑡,𝑒,)   𝐷(𝑤,𝑡,𝑒,𝑓,𝑔,)   𝑃(𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,,𝑖)   𝑇(𝑒)   𝑈(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,,𝑖)   𝐸(𝑡,𝑒,)   𝐹(𝑥,𝑦,𝑤,𝑡,𝑒,,𝑖)   𝐽(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,,𝑖)   𝑀(𝑦,𝑒)   𝑉(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,,𝑖)   𝑊(𝑥,𝑦,𝑡,𝑒,)   𝑌(𝑥,𝑦,𝑡,𝑒,)   𝑍(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,,𝑖)

Proof of Theorem stoweidlem54
StepHypRef Expression
1 stoweidlem54.3 . . 3 𝑦𝜑
2 nfv 1828 . . 3 𝑦𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
3 stoweidlem54.18 . . 3 (𝜑 → ∃𝑦(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))))
4 stoweidlem54.1 . . . . 5 𝑖𝜑
5 nfv 1828 . . . . . 6 𝑖 𝑦:(1...𝑀)⟶𝑌
6 nfra1 2920 . . . . . 6 𝑖𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
75, 6nfan 1814 . . . . 5 𝑖(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
84, 7nfan 1814 . . . 4 𝑖(𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))))
9 stoweidlem54.2 . . . . 5 𝑡𝜑
10 nfcv 2746 . . . . . . 7 𝑡𝑦
11 nfcv 2746 . . . . . . 7 𝑡(1...𝑀)
12 stoweidlem54.6 . . . . . . . 8 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
13 nfra1 2920 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
14 nfcv 2746 . . . . . . . . 9 𝑡𝐴
1513, 14nfrab 3095 . . . . . . . 8 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
1612, 15nfcxfr 2744 . . . . . . 7 𝑡𝑌
1710, 11, 16nff 5936 . . . . . 6 𝑡 𝑦:(1...𝑀)⟶𝑌
18 nfra1 2920 . . . . . . . 8 𝑡𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀)
19 nfra1 2920 . . . . . . . 8 𝑡𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)
2018, 19nfan 1814 . . . . . . 7 𝑡(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
2111, 20nfral 2924 . . . . . 6 𝑡𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
2217, 21nfan 1814 . . . . 5 𝑡(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
239, 22nfan 1814 . . . 4 𝑡(𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))))
24 stoweidlem54.4 . . . . 5 𝑤𝜑
25 nfv 1828 . . . . 5 𝑤(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
2624, 25nfan 1814 . . . 4 𝑤(𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))))
27 stoweidlem54.10 . . . . 5 𝑉 = {𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
28 nfrab1 3094 . . . . 5 𝑤{𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
2927, 28nfcxfr 2744 . . . 4 𝑤𝑉
30 stoweidlem54.7 . . . 4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
31 eqid 2605 . . . 4 (seq1(𝑃, 𝑦)‘𝑀) = (seq1(𝑃, 𝑦)‘𝑀)
32 stoweidlem54.8 . . . 4 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑦𝑖)‘𝑡)))
33 stoweidlem54.9 . . . 4 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
34 stoweidlem54.13 . . . . 5 (𝜑𝑀 ∈ ℕ)
3534adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝑀 ∈ ℕ)
36 stoweidlem54.14 . . . . 5 (𝜑𝑊:(1...𝑀)⟶𝑉)
3736adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝑊:(1...𝑀)⟶𝑉)
38 simprl 789 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝑦:(1...𝑀)⟶𝑌)
39 simpr 475 . . . . 5 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑤𝑉) → 𝑤𝑉)
4027rabeq2i 3165 . . . . . 6 (𝑤𝑉 ↔ (𝑤𝐽 ∧ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))))
4140simplbi 474 . . . . 5 (𝑤𝑉𝑤𝐽)
42 elssuni 4393 . . . . . 6 (𝑤𝐽𝑤 𝐽)
43 stoweidlem54.5 . . . . . 6 𝑇 = 𝐽
4442, 43syl6sseqr 3610 . . . . 5 (𝑤𝐽𝑤𝑇)
4539, 41, 443syl 18 . . . 4 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑤𝑉) → 𝑤𝑇)
46 stoweidlem54.16 . . . . 5 (𝜑𝐷 ran 𝑊)
4746adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝐷 ran 𝑊)
48 stoweidlem54.17 . . . . 5 (𝜑𝐷𝑇)
4948adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝐷𝑇)
50 stoweidlem54.15 . . . . 5 (𝜑𝐵𝑇)
5150adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝐵𝑇)
52 r19.26 3041 . . . . . . 7 (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)) ↔ (∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑖 ∈ (1...𝑀)∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
5352simplbi 474 . . . . . 6 (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)) → ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀))
5453ad2antll 760 . . . . 5 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀))
5554r19.21bi 2911 . . . 4 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀))
5652simprbi 478 . . . . . 6 (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)) → ∀𝑖 ∈ (1...𝑀)∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
5756ad2antll 760 . . . . 5 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → ∀𝑖 ∈ (1...𝑀)∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
5857r19.21bi 2911 . . . 4 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
59 stoweidlem54.11 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
60593adant1r 1310 . . . 4 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
61 stoweidlem54.12 . . . . 5 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
6261adantlr 746 . . . 4 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑓𝐴) → 𝑓:𝑇⟶ℝ)
63 stoweidlem54.19 . . . . 5 (𝜑𝑇 ∈ V)
6463adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝑇 ∈ V)
65 stoweidlem54.20 . . . . 5 (𝜑𝐸 ∈ ℝ+)
6665adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝐸 ∈ ℝ+)
67 stoweidlem54.21 . . . . 5 (𝜑𝐸 < (1 / 3))
6867adantr 479 . . . 4 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → 𝐸 < (1 / 3))
698, 23, 26, 29, 12, 30, 31, 32, 33, 35, 37, 38, 45, 47, 49, 51, 55, 58, 60, 62, 64, 66, 68stoweidlem51 38744 . . 3 ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
701, 2, 3, 69exlimdd 2072 . 2 (𝜑 → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
71 df-rex 2897 . 2 (∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)) ↔ ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
7270, 71sylibr 222 1 (𝜑 → ∃𝑥𝐴 (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wex 1694  wnf 1698  wcel 1975  wral 2891  wrex 2892  {crab 2895  Vcvv 3168  cdif 3532  wss 3535   cuni 4362   class class class wbr 4573  cmpt 4633  ran crn 5025  wf 5782  cfv 5786  (class class class)co 6523  cmpt2 6525  cr 9787  0cc0 9788  1c1 9789   · cmul 9793   < clt 9926  cle 9927  cmin 10113   / cdiv 10529  cn 10863  3c3 10914  +crp 11660  ...cfz 12148  seqcseq 12614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-3 10923  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-seq 12615  df-exp 12674
This theorem is referenced by:  stoweidlem57  38750
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