Theorem stoweidlem54 46045

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	⊢ Ⅎ𝑖𝜑
stoweidlem54.2	⊢ Ⅎ𝑡𝜑
stoweidlem54.3	⊢ Ⅎ𝑦𝜑
stoweidlem54.4	⊢ Ⅎ𝑤𝜑
stoweidlem54.5	⊢ 𝑇 = ∪ 𝐽
stoweidlem54.6	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem54.7	⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
stoweidlem54.8	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑦‘𝑖)‘𝑡)))
stoweidlem54.9	⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
stoweidlem54.10	⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
stoweidlem54.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem54.12	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem54.13	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem54.14	⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
stoweidlem54.15	⊢ (𝜑 → 𝐵 ⊆ 𝑇)
stoweidlem54.16	⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
stoweidlem54.17	⊢ (𝜑 → 𝐷 ⊆ 𝑇)
stoweidlem54.18	⊢ (𝜑 → ∃𝑦(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))))
stoweidlem54.19	⊢ (𝜑 → 𝑇 ∈ V)
stoweidlem54.20	⊢ (𝜑 → 𝐸 ∈ ℝ+)
stoweidlem54.21	⊢ (𝜑 → 𝐸 < (1 / 3))

Assertion

Ref	Expression
stoweidlem54	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))

Distinct variable groups:   𝑓,𝑔,ℎ,𝑖,𝑡,𝑦,𝑇   𝐴,𝑓,𝑔,ℎ,𝑡,𝑦   𝐵,𝑓,𝑔,𝑖,𝑦   𝑓,𝐸,𝑔,𝑖,𝑦   𝑓,𝐹,𝑔   𝑓,𝑀,𝑔,ℎ,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,ℎ,𝑖)   𝐴(𝑤,𝑒,𝑖)   𝐵(𝑡,𝑒,ℎ)   𝐷(𝑤,𝑡,𝑒,𝑓,𝑔,ℎ)   𝑃(𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖)   𝑇(𝑒)   𝑈(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖)   𝐸(𝑡,𝑒,ℎ)   𝐹(𝑥,𝑦,𝑤,𝑡,𝑒,ℎ,𝑖)   𝐽(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖)   𝑀(𝑦,𝑒)   𝑉(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖)   𝑊(𝑥,𝑦,𝑡,𝑒,ℎ)   𝑌(𝑥,𝑦,𝑡,𝑒,ℎ)   𝑍(𝑥,𝑦,𝑤,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖)

Proof of Theorem stoweidlem54

Step	Hyp	Ref	Expression
1		stoweidlem54.3	. . 3 ⊢ Ⅎ𝑦𝜑
2		nfv 1914	. . 3 ⊢ Ⅎ𝑦∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))
3		stoweidlem54.18	. . 3 ⊢ (𝜑 → ∃𝑦(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))))
4		stoweidlem54.1	. . . . 5 ⊢ Ⅎ𝑖𝜑
5		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑖 𝑦:(1...𝑀)⟶𝑌
6		nfra1 3253	. . . . . 6 ⊢ Ⅎ𝑖∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
7	5, 6	nfan 1899	. . . . 5 ⊢ Ⅎ𝑖(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))
8	4, 7	nfan 1899	. . . 4 ⊢ Ⅎ𝑖(𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))))
9		stoweidlem54.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
10		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑡𝑦
11		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑡(1...𝑀)
12		stoweidlem54.6	. . . . . . . 8 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
13		nfra1 3253	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
14		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐴
15	13, 14	nfrabw 3432	. . . . . . . 8 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
16	12, 15	nfcxfr 2889	. . . . . . 7 ⊢ Ⅎ𝑡𝑌
17	10, 11, 16	nff 6648	. . . . . 6 ⊢ Ⅎ𝑡 𝑦:(1...𝑀)⟶𝑌
18		nfra1 3253	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀)
19		nfra1 3253	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)
20	18, 19	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
21	11, 20	nfralw 3276	. . . . . 6 ⊢ Ⅎ𝑡∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
22	17, 21	nfan 1899	. . . . 5 ⊢ Ⅎ𝑡(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))
23	9, 22	nfan 1899	. . . 4 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))))
24		stoweidlem54.4	. . . . 5 ⊢ Ⅎ𝑤𝜑
25		nfv 1914	. . . . 5 ⊢ Ⅎ𝑤(𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))
26	24, 25	nfan 1899	. . . 4 ⊢ Ⅎ𝑤(𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))))
27		stoweidlem54.10	. . . . 5 ⊢ 𝑉 = {𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
28		nfrab1 3415	. . . . 5 ⊢ Ⅎ𝑤{𝑤 ∈ 𝐽 ∣ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))}
29	27, 28	nfcxfr 2889	. . . 4 ⊢ Ⅎ𝑤𝑉
30		stoweidlem54.7	. . . 4 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
31		eqid 2729	. . . 4 ⊢ (seq1(𝑃, 𝑦)‘𝑀) = (seq1(𝑃, 𝑦)‘𝑀)
32		stoweidlem54.8	. . . 4 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑦‘𝑖)‘𝑡)))
33		stoweidlem54.9	. . . 4 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
34		stoweidlem54.13	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
35	34	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝑀 ∈ ℕ)
36		stoweidlem54.14	. . . . 5 ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
37	36	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝑊:(1...𝑀)⟶𝑉)
38		simprl 770	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝑦:(1...𝑀)⟶𝑌)
39		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) ∧ 𝑤 ∈ 𝑉) → 𝑤 ∈ 𝑉)
40	27	reqabi 3418	. . . . . 6 ⊢ (𝑤 ∈ 𝑉 ↔ (𝑤 ∈ 𝐽 ∧ ∀𝑒 ∈ ℝ+ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))))
41	40	simplbi 497	. . . . 5 ⊢ (𝑤 ∈ 𝑉 → 𝑤 ∈ 𝐽)
42		elssuni 4888	. . . . . 6 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
43		stoweidlem54.5	. . . . . 6 ⊢ 𝑇 = ∪ 𝐽
44	42, 43	sseqtrrdi 3977	. . . . 5 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ 𝑇)
45	39, 41, 44	3syl 18	. . . 4 ⊢ (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇)
46		stoweidlem54.16	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
47	46	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝐷 ⊆ ∪ ran 𝑊)
48		stoweidlem54.17	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
49	48	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝐷 ⊆ 𝑇)
50		stoweidlem54.15	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
51	50	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝐵 ⊆ 𝑇)
52		r19.26 3089	. . . . . . 7 ⊢ (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)) ↔ (∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))
53	52	simplbi 497	. . . . . 6 ⊢ (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)) → ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀))
54	53	ad2antll 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀))
55	54	r19.21bi 3221	. . . 4 ⊢ (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀))
56	52	simprbi 496	. . . . . 6 ⊢ (∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)) → ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
57	56	ad2antll 729	. . . . 5 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → ∀𝑖 ∈ (1...𝑀)∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
58	57	r19.21bi 3221	. . . 4 ⊢ (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
59		stoweidlem54.11	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
60	59	3adant1r 1178	. . . 4 ⊢ (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
61		stoweidlem54.12	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
62	61	adantlr 715	. . . 4 ⊢ (((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
63		stoweidlem54.19	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ V)
64	63	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝑇 ∈ V)
65		stoweidlem54.20	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
66	65	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝐸 ∈ ℝ+)
67		stoweidlem54.21	. . . . 5 ⊢ (𝜑 → 𝐸 < (1 / 3))
68	67	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → 𝐸 < (1 / 3))
69	8, 23, 26, 29, 12, 30, 31, 32, 33, 35, 37, 38, 45, 47, 49, 51, 55, 58, 60, 62, 64, 66, 68	stoweidlem51 46042	. . 3 ⊢ ((𝜑 ∧ (𝑦:(1...𝑀)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑀)(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)))) → ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))
70	1, 2, 3, 69	exlimdd 2221	. 2 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))
71		df-rex 3054	. 2 ⊢ (∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))
72	70, 71	sylibr 234	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3394  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∪ cuni 4858   class class class wbr 5092   ↦ cmpt 5173  ran crn 5620  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  ℝcr 11008  0cc0 11009  1c1 11010   · cmul 11014   < clt 11149   ≤ cle 11150   − cmin 11347   / cdiv 11777  ℕcn 12128  3c3 12184  ℝ⁺crp 12893  ...cfz 13410  seqcseq 13908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-div 11778  df-nn 12129  df-2 12191  df-3 12192  df-n0 12385  df-z 12472  df-uz 12736  df-rp 12894  df-fz 13411  df-fzo 13558  df-seq 13909  df-exp 13969

This theorem is referenced by:  stoweidlem57  46048