stoweidlem57 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem57 45978

Description: There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. In this theorem, it is proven the non-trivial case (the closed set D is nonempty). Here D is used to represent A in the paper, because the variable A is used for the subalgebra of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem57.1	⊢ Ⅎ𝑡𝐷
stoweidlem57.2	⊢ Ⅎ𝑡𝑈
stoweidlem57.3	⊢ Ⅎ𝑡𝜑
stoweidlem57.4	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem57.5	⊢ 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))}
stoweidlem57.6	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem57.7	⊢ 𝑇 = ∪ 𝐽
stoweidlem57.8	⊢ 𝐶 = (𝐽 Cn 𝐾)
stoweidlem57.9	⊢ 𝑈 = (𝑇 ∖ 𝐵)
stoweidlem57.10	⊢ (𝜑 → 𝐽 ∈ Comp)
stoweidlem57.11	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
stoweidlem57.12	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem57.13	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem57.14	⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
stoweidlem57.15	⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
stoweidlem57.16	⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
stoweidlem57.17	⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽))
stoweidlem57.18	⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)
stoweidlem57.19	⊢ (𝜑 → 𝐷 ≠ ∅)
stoweidlem57.20	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem57.21	⊢ (𝜑 → 𝐸 < (1 / 3))

**Assertion**
Ref	Expression
stoweidlem57	⊢ (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: 𝑒,𝑎,𝑓,𝑡 𝑞,𝑎,𝑟,𝑓,𝑡,𝐴 𝐴,𝑒,𝑓,𝑡 𝐷,𝑎,𝑒,𝑓 𝑇,𝑎,𝑒,𝑓,𝑡 𝑈,𝑎,𝑒,𝑓 𝜑,𝑎,𝑒,𝑓 𝑒,𝑔,ℎ,𝑓,𝑡,𝐴 𝑤,𝑒,ℎ,𝑡,𝐴 𝑒,𝐸,𝑓,𝑔,ℎ,𝑡 𝑔,𝑟,ℎ,𝐴 𝑥,𝑓,𝑔,ℎ,𝑡,𝐴 𝐵,𝑓,𝑔,𝑟 𝑓,𝑉,𝑔,𝑟 𝑓,𝑌,𝑔,𝑟 𝑔,𝑞,𝐷 𝐷,ℎ,𝑟 𝑔,𝐽,ℎ,𝑡 𝑇,𝑔,ℎ,𝑟 𝑈,𝑔,ℎ,𝑟 𝜑,𝑔,ℎ,𝑟 𝑤,𝑟,𝐸 𝐴,𝑞 𝐷,𝑞 𝑇,𝑞 𝑈,𝑞 𝜑,𝑞 𝑤,𝐷 𝑤,𝐵 𝑡,𝐾 𝜑,𝑤 𝑤,𝐽 𝑤,𝑇 𝑤,𝑈 𝑤,𝑌 𝑥,𝐵 𝑥,𝐷 𝑥,𝐸 𝑥,𝑇 Allowed substitution hints: 𝜑(𝑥,𝑡) 𝐵(𝑡,𝑒,ℎ,𝑞,𝑎) 𝐶(𝑥,𝑤,𝑡,𝑒,𝑓,𝑔,ℎ,𝑟,𝑞,𝑎) 𝐷(𝑡) 𝑈(𝑥,𝑡) 𝐸(𝑞,𝑎) 𝐽(𝑥,𝑒,𝑓,𝑟,𝑞,𝑎) 𝐾(𝑥,𝑤,𝑒,𝑓,𝑔,ℎ,𝑟,𝑞,𝑎) 𝑉(𝑥,𝑤,𝑡,𝑒,ℎ,𝑞,𝑎) 𝑌(𝑥,𝑡,𝑒,ℎ,𝑞,𝑎)

Proof of Theorem stoweidlem57 Dummy variables 𝑠 𝑚 𝑖 𝑣 𝑦 𝑢 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem57.2	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑈
2		stoweidlem57.3	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝜑
3		stoweidlem57.1	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝐷
4	3	nfcri 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑠 ∈ 𝐷
5	2, 4	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑠 ∈ 𝐷)
6		stoweidlem57.6	. . . . . . . . . 10 ⊢ 𝐾 = (topGen‘ran (,))
7		stoweidlem57.10	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Comp)
8	7	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝐽 ∈ Comp)
9		stoweidlem57.7	. . . . . . . . . 10 ⊢ 𝑇 = ∪ 𝐽
10		stoweidlem57.8	. . . . . . . . . 10 ⊢ 𝐶 = (𝐽 Cn 𝐾)
11		stoweidlem57.11	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
12	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝐴 ⊆ 𝐶)
13		stoweidlem57.12	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
14	13	3adant1r 1177	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐷) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
15		stoweidlem57.13	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
16	15	3adant1r 1177	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐷) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
17		stoweidlem57.14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
18	17	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐷) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
19		stoweidlem57.15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
20	19	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝐷) ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
21		stoweidlem57.9	. . . . . . . . . . . 12 ⊢ 𝑈 = (𝑇 ∖ 𝐵)
22		stoweidlem57.16	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ (Clsd‘𝐽))
23		cmptop 23424	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
24	9	iscld 23056	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝐵 ⊆ 𝑇 ∧ (𝑇 ∖ 𝐵) ∈ 𝐽)))
25	7, 23, 24	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝐵 ⊆ 𝑇 ∧ (𝑇 ∖ 𝐵) ∈ 𝐽)))
26	22, 25	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ⊆ 𝑇 ∧ (𝑇 ∖ 𝐵) ∈ 𝐽))
27	26	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∖ 𝐵) ∈ 𝐽)
28	21, 27	eqeltrid 2848	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
29	28	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑈 ∈ 𝐽)
30		stoweidlem57.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽))
31	9	cldss 23058	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ 𝑇)
32	30, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
33	32	sselda 4008	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ 𝑇)
34		stoweidlem57.18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)
35		disjr 4474	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∩ 𝐷) = ∅ ↔ ∀𝑠 ∈ 𝐷 ¬ 𝑠 ∈ 𝐵)
36	34, 35	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑠 ∈ 𝐷 ¬ 𝑠 ∈ 𝐵)
37	36	r19.21bi 3257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → ¬ 𝑠 ∈ 𝐵)
38	33, 37	eldifd 3987	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝑇 ∖ 𝐵))
39	38, 21	eleqtrrdi 2855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ 𝑈)
40	1, 5, 6, 8, 9, 10, 12, 14, 16, 18, 20, 29, 39	stoweidlem56 45977	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → ∃𝑤 ∈ 𝐽 ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀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		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐽 ∧ ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))) → 𝑤 ∈ 𝐽)
42		simprll 778	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐽 ∧ ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))) → 𝑠 ∈ 𝑤)
43		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐽 ∧ ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))) → ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))
44		stoweidlem57.5	. . . . . . . . . . . . 13 ⊢ 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))}
45	44	reqabi 3467	. . . . . . . . . . . 12 ⊢ (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))))
46	41, 43, 45	sylanbrc 582	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝐽 ∧ ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))) → 𝑤 ∈ 𝑉)
47	41, 42, 46	jca32 515	. . . . . . . . . 10 ⊢ ((𝑤 ∈ 𝐽 ∧ ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)))) → (𝑤 ∈ 𝐽 ∧ (𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉)))
48	47	reximi2 3085	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐽 ((𝑠 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))) → ∃𝑤 ∈ 𝐽 (𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
49		rexex 3082	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐽 (𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
50	40, 48, 49	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
51		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑠
52		nfrab1 3464	. . . . . . . . . 10 ⊢ 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))}
53	44, 52	nfcxfr 2906	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑉
54	51, 53	elunif 44916	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ 𝑉 ↔ ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
55	50, 54	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ ∪ 𝑉)
56	55	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ 𝐷 → 𝑠 ∈ ∪ 𝑉))
57	56	ssrdv 4014	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑉)
58		cmpcld 23431	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ 𝐷 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝐷) ∈ Comp)
59	7, 30, 58	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾_t 𝐷) ∈ Comp)
60	7, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Top)
61	9	cmpsub 23429	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐷 ⊆ 𝑇) → ((𝐽 ↾_t 𝐷) ∈ Comp ↔ ∀𝑘 ∈ 𝒫 𝐽(𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢)))
62	60, 32, 61	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝐷) ∈ Comp ↔ ∀𝑘 ∈ 𝒫 𝐽(𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢)))
63	59, 62	mpbid 232	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝒫 𝐽(𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
64		ssrab2 4103	. . . . . . . 8 ⊢ {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))} ⊆ 𝐽
65	44, 64	eqsstri 4043	. . . . . . 7 ⊢ 𝑉 ⊆ 𝐽
66	44, 7	rabexd 5358	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V)
67		elpwg 4625	. . . . . . . 8 ⊢ (𝑉 ∈ V → (𝑉 ∈ 𝒫 𝐽 ↔ 𝑉 ⊆ 𝐽))
68	66, 67	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝒫 𝐽 ↔ 𝑉 ⊆ 𝐽))
69	65, 68	mpbiri 258	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝒫 𝐽)
70		unieq 4942	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → ∪ 𝑘 = ∪ 𝑉)
71	70	sseq2d 4041	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (𝐷 ⊆ ∪ 𝑘 ↔ 𝐷 ⊆ ∪ 𝑉))
72		pweq 4636	. . . . . . . . . 10 ⊢ (𝑘 = 𝑉 → 𝒫 𝑘 = 𝒫 𝑉)
73	72	ineq1d 4240	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → (𝒫 𝑘 ∩ Fin) = (𝒫 𝑉 ∩ Fin))
74	73	rexeqdv 3335	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢 ↔ ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
75	71, 74	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = 𝑉 → ((𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢) ↔ (𝐷 ⊆ ∪ 𝑉 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢)))
76	75	rspccva 3634	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝒫 𝐽(𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢) ∧ 𝑉 ∈ 𝒫 𝐽) → (𝐷 ⊆ ∪ 𝑉 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
77	63, 69, 76	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑉 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
78	57, 77	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢)
79		elinel1 4224	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → 𝑢 ∈ 𝒫 𝑉)
80		elpwi 4629	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝒫 𝑉 → 𝑢 ⊆ 𝑉)
81	80	ssdifssd 4170	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 𝑉 → (𝑢 ∖ {∅}) ⊆ 𝑉)
82		vex 3492	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
83		difexg 5347	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ V → (𝑢 ∖ {∅}) ∈ V)
84	82, 83	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑢 ∖ {∅}) ∈ V
85	84	elpw 4626	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {∅}) ∈ 𝒫 𝑉 ↔ (𝑢 ∖ {∅}) ⊆ 𝑉)
86	81, 85	sylibr 234	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑉 → (𝑢 ∖ {∅}) ∈ 𝒫 𝑉)
87	79, 86	syl 17	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ 𝒫 𝑉)
88		elinel2 4225	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → 𝑢 ∈ Fin)
89		diffi 9242	. . . . . . . . 9 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {∅}) ∈ Fin)
90	88, 89	syl 17	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ Fin)
91	87, 90	elind 4223	. . . . . . 7 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin))
92	91	3ad2ant2 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → (𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin))
93		unidif0 5378	. . . . . . . . 9 ⊢ ∪ (𝑢 ∖ {∅}) = ∪ 𝑢
94	93	sseq2i 4038	. . . . . . . 8 ⊢ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ↔ 𝐷 ⊆ ∪ 𝑢)
95	94	biimpri 228	. . . . . . 7 ⊢ (𝐷 ⊆ ∪ 𝑢 → 𝐷 ⊆ ∪ (𝑢 ∖ {∅}))
96	95	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → 𝐷 ⊆ ∪ (𝑢 ∖ {∅}))
97		eldifsni 4815	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑢 ∖ {∅}) → 𝑤 ≠ ∅)
98	97	rgen 3069	. . . . . . 7 ⊢ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅
99	98	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)
100		unieq 4942	. . . . . . . . 9 ⊢ (𝑟 = (𝑢 ∖ {∅}) → ∪ 𝑟 = ∪ (𝑢 ∖ {∅}))
101	100	sseq2d 4041	. . . . . . . 8 ⊢ (𝑟 = (𝑢 ∖ {∅}) → (𝐷 ⊆ ∪ 𝑟 ↔ 𝐷 ⊆ ∪ (𝑢 ∖ {∅})))
102		raleq 3331	. . . . . . . 8 ⊢ (𝑟 = (𝑢 ∖ {∅}) → (∀𝑤 ∈ 𝑟 𝑤 ≠ ∅ ↔ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅))
103	101, 102	anbi12d 631	. . . . . . 7 ⊢ (𝑟 = (𝑢 ∖ {∅}) → ((𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅) ↔ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ∧ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)))
104	103	rspcev 3635	. . . . . 6 ⊢ (((𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ∧ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)) → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
105	92, 96, 99, 104	syl12anc 836	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
106	105	rexlimdv3a 3165	. . . 4 ⊢ (𝜑 → (∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢 → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)))
107	78, 106	mpd 15	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
108		nfv 1913	. . . . . 6 ⊢ Ⅎℎ𝜑
109		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎℎℝ⁺
110		nfre1 3291	. . . . . . . . . . . 12 ⊢ Ⅎℎ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))
111	109, 110	nfralw 3317	. . . . . . . . . . 11 ⊢ Ⅎℎ∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))
112		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎℎ𝐽
113	111, 112	nfrabw 3483	. . . . . . . . . 10 ⊢ Ⅎℎ{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))}
114	44, 113	nfcxfr 2906	. . . . . . . . 9 ⊢ Ⅎℎ𝑉
115	114	nfpw 4641	. . . . . . . 8 ⊢ Ⅎℎ𝒫 𝑉
116		nfcv 2908	. . . . . . . 8 ⊢ ℲℎFin
117	115, 116	nfin 4245	. . . . . . 7 ⊢ Ⅎℎ(𝒫 𝑉 ∩ Fin)
118	117	nfcri 2900	. . . . . 6 ⊢ Ⅎℎ 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
119		nfv 1913	. . . . . 6 ⊢ Ⅎℎ(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
120	108, 118, 119	nf3an 1900	. . . . 5 ⊢ Ⅎℎ(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
121		nfcv 2908	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡ℝ⁺
122		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝐴
123		nfra1 3290	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
124		nfra1 3290	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒
125		nfra1 3290	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)
126	123, 124, 125	nf3an 1900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))
127	122, 126	nfrexw 3319	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))
128	121, 127	nfralw 3317	. . . . . . . . . . 11 ⊢ Ⅎ𝑡∀𝑒 ∈ ℝ⁺ ∃ℎ ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡))
129		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐽
130	128, 129	nfrabw 3483	. . . . . . . . . 10 ⊢ Ⅎ𝑡{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))}
131	44, 130	nfcxfr 2906	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑉
132	131	nfpw 4641	. . . . . . . 8 ⊢ Ⅎ𝑡𝒫 𝑉
133		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑡Fin
134	132, 133	nfin 4245	. . . . . . 7 ⊢ Ⅎ𝑡(𝒫 𝑉 ∩ Fin)
135	134	nfcri 2900	. . . . . 6 ⊢ Ⅎ𝑡 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
136		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑡∪ 𝑟
137	3, 136	nfss 4001	. . . . . . 7 ⊢ Ⅎ𝑡 𝐷 ⊆ ∪ 𝑟
138		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑡∀𝑤 ∈ 𝑟 𝑤 ≠ ∅
139	137, 138	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑡(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
140	2, 135, 139	nf3an 1900	. . . . 5 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
141		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑤𝜑
142	53	nfpw 4641	. . . . . . . 8 ⊢ Ⅎ𝑤𝒫 𝑉
143		nfcv 2908	. . . . . . . 8 ⊢ Ⅎ𝑤Fin
144	142, 143	nfin 4245	. . . . . . 7 ⊢ Ⅎ𝑤(𝒫 𝑉 ∩ Fin)
145	144	nfcri 2900	. . . . . 6 ⊢ Ⅎ𝑤 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
146		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑤 𝐷 ⊆ ∪ 𝑟
147		nfra1 3290	. . . . . . 7 ⊢ Ⅎ𝑤∀𝑤 ∈ 𝑟 𝑤 ≠ ∅
148	146, 147	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑤(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
149	141, 145, 148	nf3an 1900	. . . . 5 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
150		stoweidlem57.4	. . . . 5 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
151		simp2 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝑟 ∈ (𝒫 𝑉 ∩ Fin))
152		simp3l 1201	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐷 ⊆ ∪ 𝑟)
153		stoweidlem57.19	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ ∅)
154	153	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐷 ≠ ∅)
155		stoweidlem57.20	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
156	155	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐸 ∈ ℝ⁺)
157	26	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
158	157	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐵 ⊆ 𝑇)
159	66	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝑉 ∈ V)
160		retop 24803	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
161	6, 160	eqeltri 2840	. . . . . . . 8 ⊢ 𝐾 ∈ Top
162		cnfex 44928	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
163	60, 161, 162	sylancl 585	. . . . . . 7 ⊢ (𝜑 → (𝐽 Cn 𝐾) ∈ V)
164	11, 10	sseqtrdi 4059	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
165	163, 164	ssexd 5342	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
166	165	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐴 ∈ V)
167	120, 140, 149, 21, 150, 44, 151, 152, 154, 156, 158, 159, 166	stoweidlem39 45960	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))))
168	167	rexlimdv3a 3165	. . 3 ⊢ (𝜑 → (∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅) → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))))
169	107, 168	mpd 15	. 2 ⊢ (𝜑 → ∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))))
170		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑚 ∈ ℕ)
171		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑖 𝑣:(1...𝑚)⟶𝑉
172		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐷 ⊆ ∪ ran 𝑣
173		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑦:(1...𝑚)⟶𝑌
174		nfra1 3290	. . . . . . . . . 10 ⊢ Ⅎ𝑖∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))
175	173, 174	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑖(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))
176	175	nfex 2328	. . . . . . . 8 ⊢ Ⅎ𝑖∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))
177	171, 172, 176	nf3an 1900	. . . . . . 7 ⊢ Ⅎ𝑖(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
178	170, 177	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑖((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))))
179		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
180	2, 179	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
181		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑣
182		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑡(1...𝑚)
183	181, 182, 131	nff 6743	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)⟶𝑉
184		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑡∪ ran 𝑣
185	3, 184	nfss 4001	. . . . . . . 8 ⊢ Ⅎ𝑡 𝐷 ⊆ ∪ ran 𝑣
186		nfcv 2908	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑦
187	123, 122	nfrabw 3483	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
188	150, 187	nfcxfr 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑌
189	186, 182, 188	nff 6743	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑦:(1...𝑚)⟶𝑌
190		nfra1 3290	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚)
191		nfra1 3290	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)
192	190, 191	nfan 1898	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))
193	182, 192	nfralw 3317	. . . . . . . . . 10 ⊢ Ⅎ𝑡∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))
194	189, 193	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))
195	194	nfex 2328	. . . . . . . 8 ⊢ Ⅎ𝑡∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))
196	183, 185, 195	nf3an 1900	. . . . . . 7 ⊢ Ⅎ𝑡(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
197	180, 196	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))))
198		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑚 ∈ ℕ)
199		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣:(1...𝑚)⟶𝑉
200		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ ran 𝑣
201		nfe1 2151	. . . . . . . 8 ⊢ Ⅎ𝑦∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))
202	199, 200, 201	nf3an 1900	. . . . . . 7 ⊢ Ⅎ𝑦(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
203	198, 202	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑦((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))))
204		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
205		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑣
206		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑤(1...𝑚)
207	205, 206, 53	nff 6743	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)⟶𝑉
208		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑤 𝐷 ⊆ ∪ ran 𝑣
209		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑤∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))
210	207, 208, 209	nf3an 1900	. . . . . . 7 ⊢ Ⅎ𝑤(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
211	204, 210	nfan 1898	. . . . . 6 ⊢ Ⅎ𝑤((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))))
212		eqid 2740	. . . . . 6 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
213		eqid 2740	. . . . . 6 ⊢ (𝑓 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}, 𝑔 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))) = (𝑓 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}, 𝑔 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
214		eqid 2740	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡)))
215		eqid 2740	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (seq1( · , ((𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡)))‘𝑡))‘𝑚)) = (𝑡 ∈ 𝑇 ↦ (seq1( · , ((𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡)))‘𝑡))‘𝑚))
216		simp1ll 1236	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → 𝜑)
217	216, 15	syld3an1 1410	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘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)
218	11	sselda 4008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
219	6, 9, 10, 218	fcnre 44925	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
220	219	ad4ant14 751	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
221		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝑚 ∈ ℕ)
222		simpr1 1194	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝑣:(1...𝑚)⟶𝑉)
223	9	cldss 23058	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
224	22, 223	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
225	224	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝐵 ⊆ 𝑇)
226		simpr2 1195	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝐷 ⊆ ∪ ran 𝑣)
227	32	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝐷 ⊆ 𝑇)
228		feq3 6730	. . . . . . . . . . . 12 ⊢ (𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} → (𝑦:(1...𝑚)⟶𝑌 ↔ 𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}))
229	150, 228	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑦:(1...𝑚)⟶𝑌 ↔ 𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
230	229	biimpi 216	. . . . . . . . . 10 ⊢ (𝑦:(1...𝑚)⟶𝑌 → 𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
231	230	anim1i 614	. . . . . . . . 9 ⊢ ((𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))) → (𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
232	231	eximi 1833	. . . . . . . 8 ⊢ (∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))) → ∃𝑦(𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
233	232	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))) → ∃𝑦(𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
234	233	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → ∃𝑦(𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))
235	7	uniexd 7777	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
236	9, 235	eqeltrid 2848	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
237	236	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝑇 ∈ V)
238	155	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝐸 ∈ ℝ⁺)
239		stoweidlem57.21	. . . . . . 7 ⊢ (𝜑 → 𝐸 < (1 / 3))
240	239	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → 𝐸 < (1 / 3))
241	178, 197, 203, 211, 9, 212, 213, 214, 215, 44, 217, 220, 221, 222, 225, 226, 227, 234, 237, 238, 240	stoweidlem54 45975	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))))) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))
242	241	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))
243	242	exlimdv 1932	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (∃𝑣(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))
244	243	rexlimdva 3161	. 2 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑣(𝑣:(1...𝑚)⟶𝑉 ∧ 𝐷 ⊆ ∪ ran 𝑣 ∧ ∃𝑦(𝑦:(1...𝑚)⟶𝑌 ∧ ∀𝑖 ∈ (1...𝑚)(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)))) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))
245	169, 244	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∃wex 1777 Ⅎwnf 1781 ∈ wcel 2108 Ⅎwnfc 2893 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 {csn 4648 ∪ cuni 4931 class class class wbr 5166 ↦ cmpt 5249 ran crn 5701 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 Fincfn 9003 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 / cdiv 11947 ℕcn 12293 3c3 12349 ℝ⁺crp 13057 (,)cioo 13407 ...cfz 13567 seqcseq 14052 ↾_t crest 17480 topGenctg 17497 Topctop 22920 Clsdccld 23045 Cn ccn 23253 Compccmp 23415

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-cn 23256 df-cnp 23257 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-xms 24351 df-ms 24352 df-tms 24353

This theorem is referenced by: stoweidlem58 45979

W3C validator