Theorem stoweidlem54 46504

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 1916	. . 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 1916	. . . . . 6 ⊢ Ⅎ𝑖 𝑦:(1...𝑀)⟶𝑌
6		nfra1 3262	. . . . . 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 1901	. . . . 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 1901	. . . 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 2899	. . . . . . 7 ⊢ Ⅎ𝑡𝑦
11		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑡(1...𝑀)
12		stoweidlem54.6	. . . . . . . 8 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
13		nfra1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
14		nfcv 2899	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐴
15	13, 14	nfrabw 3427	. . . . . . . 8 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
16	12, 15	nfcxfr 2897	. . . . . . 7 ⊢ Ⅎ𝑡𝑌
17	10, 11, 16	nff 6660	. . . . . 6 ⊢ Ⅎ𝑡 𝑦:(1...𝑀)⟶𝑌
18		nfra1 3262	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀)
19		nfra1 3262	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)
20	18, 19	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
21	11, 20	nfralw 3285	. . . . . 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 1901	. . . . 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 1901	. . . 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 1916	. . . . 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 1901	. . . 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 3410	. . . . 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 2897	. . . 4 ⊢ Ⅎ𝑤𝑉
30		stoweidlem54.7	. . . 4 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
31		eqid 2737	. . . 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 771	. . . 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 3413	. . . . . 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 496	. . . . 5 ⊢ (𝑤 ∈ 𝑉 → 𝑤 ∈ 𝐽)
42		elssuni 4882	. . . . . 6 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
43		stoweidlem54.5	. . . . . 6 ⊢ 𝑇 = ∪ 𝐽
44	42, 43	sseqtrrdi 3964	. . . . 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 3098	. . . . . . 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 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 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀))
54	53	ad2antll 730	. . . . 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 3230	. . . 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 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 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
57	56	ad2antll 730	. . . . 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 3230	. . . 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 1179	. . . 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 716	. . . 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 46501	. . 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 2228	. 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 3063	. 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 1087   = wceq 1542  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430   ∖ cdif 3887   ⊆ wss 3890  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167  ran crn 5627  ⟶wf 6490  ‘cfv 6494  (class class class)co 7362   ∈ cmpo 7364  ℝcr 11032  0cc0 11033  1c1 11034   · cmul 11038   < clt 11174   ≤ cle 11175   − cmin 11372   / cdiv 11802  ℕcn 12169  3c3 12232  ℝ⁺crp 12937  ...cfz 13456  seqcseq 13958

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7813  df-1st 7937  df-2nd 7938  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-er 8638  df-en 8889  df-dom 8890  df-sdom 8891  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-n0 12433  df-z 12520  df-uz 12784  df-rp 12938  df-fz 13457  df-fzo 13604  df-seq 13959  df-exp 14019

This theorem is referenced by:  stoweidlem57  46507