Theorem stoweidlem54 46845

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 1947	. . 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 1947	. . . . . 6 ⊢ Ⅎ𝑖 𝑦:(1...𝑀)⟶𝑌
6		nfra1 3291	. . . . . 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 1932	. . . . 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 1932	. . . 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 2927	. . . . . . 7 ⊢ Ⅎ𝑡𝑦
11		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑡(1...𝑀)
12		stoweidlem54.6	. . . . . . . 8 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
13		nfra1 3291	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
14		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐴
15	13, 14	nfrabw 3454	. . . . . . . 8 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
16	12, 15	nfcxfr 2925	. . . . . . 7 ⊢ Ⅎ𝑡𝑌
17	10, 11, 16	nff 6705	. . . . . 6 ⊢ Ⅎ𝑡 𝑦:(1...𝑀)⟶𝑌
18		nfra1 3291	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀)
19		nfra1 3291	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡)
20	18, 19	nfan 1932	. . . . . . 7 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑊‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑀) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑦‘𝑖)‘𝑡))
21	11, 20	nfralw 3314	. . . . . 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 1932	. . . . 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 1932	. . . 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 1947	. . . . 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 1932	. . . 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 3438	. . . . 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 2925	. . . 4 ⊢ Ⅎ𝑤𝑉
30		stoweidlem54.7	. . . 4 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
31		eqid 2765	. . . 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 486	. . . 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 486	. . . 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 783	. . . 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 490	. . . . 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 3441	. . . . . 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 502	. . . . 5 ⊢ (𝑤 ∈ 𝑉 → 𝑤 ∈ 𝐽)
42		elssuni 4906	. . . . . 6 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽)
43		stoweidlem54.5	. . . . . 6 ⊢ 𝑇 = ∪ 𝐽
44	42, 43	sseqtrrdi 3979	. . . . 5 ⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ 𝑇)
45	39, 41, 44	3syl 19	. . . 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 486	. . . 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 486	. . . 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 486	. . . 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 3127	. . . . . . 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 502	. . . . . 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 742	. . . . 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 3259	. . . 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 503	. . . . . 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 742	. . . . 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 3259	. . . 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 1196	. . . 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 728	. . . 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 486	. . . 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 486	. . . 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 486	. . . 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 46842	. . 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 2259	. 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 3092	. 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 237	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∖ cdif 3903   ⊆ wss 3906  ∪ cuni 4874   class class class wbr 5111   ↦ cmpt 5194  ran crn 5664  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  ℝcr 11118  0cc0 11119  1c1 11120   · cmul 11124   < clt 11262   ≤ cle 11263   − cmin 11460   / cdiv 11890  ℕcn 12252  3c3 12315  ℝ⁺crp 13036  ...cfz 13555  seqcseq 14059

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-div 11891  df-nn 12253  df-2 12322  df-3 12323  df-n0 12524  df-z 12611  df-uz 12883  df-rp 13037  df-fz 13556  df-fzo 13704  df-seq 14060  df-exp 14120

This theorem is used by:  stoweidlem57  46848