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Theorem stoweidlem54 42693
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 1915 . . 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 1915 . . . . . 6 𝑖 𝑦:(1...𝑀)⟶𝑌
6 nfra1 3186 . . . . . 6 𝑖𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
75, 6nfan 1900 . . . . 5 𝑖(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
84, 7nfan 1900 . . . 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 2958 . . . . . . 7 𝑡𝑦
11 nfcv 2958 . . . . . . 7 𝑡(1...𝑀)
12 stoweidlem54.6 . . . . . . . 8 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
13 nfra1 3186 . . . . . . . . 9 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
14 nfcv 2958 . . . . . . . . 9 𝑡𝐴
1513, 14nfrabw 3341 . . . . . . . 8 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
1612, 15nfcxfr 2956 . . . . . . 7 𝑡𝑌
1710, 11, 16nff 6487 . . . . . 6 𝑡 𝑦:(1...𝑀)⟶𝑌
18 nfra1 3186 . . . . . . . 8 𝑡𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀)
19 nfra1 3186 . . . . . . . 8 𝑡𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)
2018, 19nfan 1900 . . . . . . 7 𝑡(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
2111, 20nfralw 3192 . . . . . 6 𝑡𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡))
2217, 21nfan 1900 . . . . 5 𝑡(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
239, 22nfan 1900 . . . 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 1915 . . . . 5 𝑤(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))
2624, 25nfan 1900 . . . 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 3340 . . . . 5 𝑤{𝑤𝐽 ∣ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))}
2927, 28nfcxfr 2956 . . . 4 𝑤𝑉
30 stoweidlem54.7 . . . 4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
31 eqid 2801 . . . 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 484 . . . 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 484 . . . 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 770 . . . 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 488 . . . . 5 (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊𝑖)((𝑦𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦𝑖)‘𝑡)))) ∧ 𝑤𝑉) → 𝑤𝑉)
4027rabeq2i 3438 . . . . . 6 (𝑤𝑉 ↔ (𝑤𝐽 ∧ ∀𝑒 ∈ ℝ+𝐴 (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ∧ ∀𝑡𝑤 (𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇𝑈)(1 − 𝑒) < (𝑡))))
4140simplbi 501 . . . . 5 (𝑤𝑉𝑤𝐽)
42 elssuni 4833 . . . . . 6 (𝑤𝐽𝑤 𝐽)
43 stoweidlem54.5 . . . . . 6 𝑇 = 𝐽
4442, 43sseqtrrdi 3969 . . . . 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 484 . . . 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 484 . . . 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 484 . . . 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 3140 . . . . . . 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 501 . . . . . 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 728 . . . . 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 3176 . . . 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 500 . . . . . 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 728 . . . . 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 3176 . . . 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 1174 . . . 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 714 . . . 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 484 . . . 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 484 . . . 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 484 . . . 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 42690 . . 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 2219 . 2 (𝜑 → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
71 df-rex 3115 . 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 237 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 399  w3a 1084   = wceq 1538  wex 1781  wnf 1785  wcel 2112  wral 3109  wrex 3110  {crab 3113  Vcvv 3444  cdif 3881  wss 3884   cuni 4803   class class class wbr 5033  cmpt 5113  ran crn 5524  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  cr 10529  0cc0 10530  1c1 10531   · cmul 10535   < clt 10668  cle 10669  cmin 10863   / cdiv 11290  cn 11629  3c3 11685  +crp 12381  ...cfz 12889  seqcseq 13368
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890  df-fzo 13033  df-seq 13369  df-exp 13430
This theorem is referenced by:  stoweidlem57  42696
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