Theorem stoweidlem54 42333

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 1911	. . 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 1911	. . . . . 6 ⊢ Ⅎ𝑖 𝑦:(1...𝑀)⟶𝑌
6		nfra1 3219	. . . . . 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 1896	. . . . 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 1896	. . . 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 2977	. . . . . . 7 ⊢ Ⅎ𝑡𝑦
11		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑡(1...𝑀)
12		stoweidlem54.6	. . . . . . . 8 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
13		nfra1 3219	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
14		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐴
15	13, 14	nfrabw 3385	. . . . . . . 8 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
16	12, 15	nfcxfr 2975	. . . . . . 7 ⊢ Ⅎ𝑡𝑌
17	10, 11, 16	nff 6504	. . . . . 6 ⊢ Ⅎ𝑡 𝑦:(1...𝑀)⟶𝑌
18		nfra1 3219	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀)
19		nfra1 3219	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)
20	18, 19	nfan 1896	. . . . . . 7 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
21	11, 20	nfralw 3225	. . . . . 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 1896	. . . . 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 1896	. . . 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 1911	. . . . 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 1896	. . . 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 3384	. . . . 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 2975	. . . 4 ⊢ Ⅎ𝑤𝑉
30		stoweidlem54.7	. . . 4 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
31		eqid 2821	. . . 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 483	. . . 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 483	. . . 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 769	. . . 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 487	. . . . 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	rabeq2i 3487	. . . . . 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 500	. . . . 5 ⊢ (𝑤 ∈ 𝑉 → 𝑤 ∈ 𝐽)
42		elssuni 4860	. . . . . 6 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
43		stoweidlem54.5	. . . . . 6 ⊢ 𝑇 = ∪ 𝐽
44	42, 43	sseqtrrdi 4017	. . . . 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 483	. . . 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 483	. . . 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 483	. . . 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 3170	. . . . . . 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 500	. . . . . 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 727	. . . . 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 3208	. . . 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 499	. . . . . 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 727	. . . . 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 3208	. . . 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 1173	. . . 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 713	. . . 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 483	. . . 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 483	. . . 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 483	. . . 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 42330	. . 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 2216	. 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 3144	. 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 236	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 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776  Ⅎwnf 1780   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ∖ cdif 3932   ⊆ wss 3935  ∪ cuni 4831   class class class wbr 5058   ↦ cmpt 5138  ran crn 5550  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152  ℝcr 10530  0cc0 10531  1c1 10532   · cmul 10536   < clt 10669   ≤ cle 10670   − cmin 10864   / cdiv 11291  ℕcn 11632  3c3 11687  ℝ⁺crp 12383  ...cfz 12886  seqcseq 13363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12384  df-fz 12887  df-fzo 13028  df-seq 13364  df-exp 13424

This theorem is referenced by:  stoweidlem57  42336