stoweidlem57 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem57 42699

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 2943	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑠 ∈ 𝐷
5	2, 4	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑠 ∈ 𝐷)
6		stoweidlem57.6	. . . . . . . . . 10 ⊢ 𝐾 = (topGen‘ran (,))
7		stoweidlem57.10	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Comp)
8	7	adantr 484	. . . . . . . . . 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 484	. . . . . . . . . 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 1174	. . . . . . . . . 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 1174	. . . . . . . . . 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 22000	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
24	9	iscld 21632	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 ⊆ 𝑇 ∧ (𝑇 ∖ 𝐵) ∈ 𝐽))
27	26	simprd 499	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∖ 𝐵) ∈ 𝐽)
28	21, 27	eqeltrid 2894	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
29	28	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑈 ∈ 𝐽)
30		stoweidlem57.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽))
31	9	cldss 21634	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ 𝑇)
32	30, 31	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
33	32	sselda 3915	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ 𝑇)
34		stoweidlem57.18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)
35		disjr 4357	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∩ 𝐷) = ∅ ↔ ∀𝑠 ∈ 𝐷 ¬ 𝑠 ∈ 𝐵)
36	34, 35	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑠 ∈ 𝐷 ¬ 𝑠 ∈ 𝐵)
37	36	r19.21bi 3173	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → ¬ 𝑠 ∈ 𝐵)
38	33, 37	eldifd 3892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝑇 ∖ 𝐵))
39	38, 21	eleqtrrdi 2901	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ 𝑈)
40	1, 5, 6, 8, 9, 10, 12, 14, 16, 18, 20, 29, 39	stoweidlem56 42698	. . . . . . . . 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 486	. . . . . . . . . . 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	rabeq2i 3435	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 519	. . . . . . . . . 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 3207	. . . . . . . . 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 3203	. . . . . . . . 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 2955	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑠
52		nfrab1 3337	. . . . . . . . . 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 2953	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑉
54	51, 53	elunif 41645	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ 𝑉 ↔ ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
55	50, 54	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ ∪ 𝑉)
56	55	ex 416	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ 𝐷 → 𝑠 ∈ ∪ 𝑉))
57	56	ssrdv 3921	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑉)
58		cmpcld 22007	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ 𝐷 ∈ (Clsd‘𝐽)) → (𝐽 ↾_t 𝐷) ∈ Comp)
59	7, 30, 58	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾_t 𝐷) ∈ Comp)
60	7, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Top)
61	9	cmpsub 22005	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → ((𝐽 ↾_t 𝐷) ∈ Comp ↔ ∀𝑘 ∈ 𝒫 𝐽(𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢)))
63	59, 62	mpbid 235	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝒫 𝐽(𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
64		ssrab2 4007	. . . . . . . 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 3949	. . . . . . 7 ⊢ 𝑉 ⊆ 𝐽
66	44, 7	rabexd 5200	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V)
67		elpwg 4500	. . . . . . . 8 ⊢ (𝑉 ∈ V → (𝑉 ∈ 𝒫 𝐽 ↔ 𝑉 ⊆ 𝐽))
68	66, 67	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝒫 𝐽 ↔ 𝑉 ⊆ 𝐽))
69	65, 68	mpbiri 261	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝒫 𝐽)
70		unieq 4811	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → ∪ 𝑘 = ∪ 𝑉)
71	70	sseq2d 3947	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (𝐷 ⊆ ∪ 𝑘 ↔ 𝐷 ⊆ ∪ 𝑉))
72		pweq 4513	. . . . . . . . . 10 ⊢ (𝑘 = 𝑉 → 𝒫 𝑘 = 𝒫 𝑉)
73	72	ineq1d 4138	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → (𝒫 𝑘 ∩ Fin) = (𝒫 𝑉 ∩ Fin))
74	73	rexeqdv 3365	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢 ↔ ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
75	71, 74	imbi12d 348	. . . . . . 7 ⊢ (𝑘 = 𝑉 → ((𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢) ↔ (𝐷 ⊆ ∪ 𝑉 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢)))
76	75	rspccva 3570	. . . . . 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 587	. . . . 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 4122	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → 𝑢 ∈ 𝒫 𝑉)
80		elpwi 4506	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝒫 𝑉 → 𝑢 ⊆ 𝑉)
81	80	ssdifssd 4070	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 𝑉 → (𝑢 ∖ {∅}) ⊆ 𝑉)
82		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
83		difexg 5195	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ V → (𝑢 ∖ {∅}) ∈ V)
84	82, 83	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑢 ∖ {∅}) ∈ V
85	84	elpw 4501	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {∅}) ∈ 𝒫 𝑉 ↔ (𝑢 ∖ {∅}) ⊆ 𝑉)
86	81, 85	sylibr 237	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑉 → (𝑢 ∖ {∅}) ∈ 𝒫 𝑉)
87	79, 86	syl 17	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ 𝒫 𝑉)
88		elinel2 4123	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → 𝑢 ∈ Fin)
89		diffi 8734	. . . . . . . . 9 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {∅}) ∈ Fin)
90	88, 89	syl 17	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ Fin)
91	87, 90	elind 4121	. . . . . . 7 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin))
92	91	3ad2ant2 1131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → (𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin))
93		unidif0 5225	. . . . . . . . 9 ⊢ ∪ (𝑢 ∖ {∅}) = ∪ 𝑢
94	93	sseq2i 3944	. . . . . . . 8 ⊢ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ↔ 𝐷 ⊆ ∪ 𝑢)
95	94	biimpri 231	. . . . . . 7 ⊢ (𝐷 ⊆ ∪ 𝑢 → 𝐷 ⊆ ∪ (𝑢 ∖ {∅}))
96	95	3ad2ant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → 𝐷 ⊆ ∪ (𝑢 ∖ {∅}))
97		eldifsni 4683	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑢 ∖ {∅}) → 𝑤 ≠ ∅)
98	97	rgen 3116	. . . . . . 7 ⊢ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅
99	98	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)
100		unieq 4811	. . . . . . . . 9 ⊢ (𝑟 = (𝑢 ∖ {∅}) → ∪ 𝑟 = ∪ (𝑢 ∖ {∅}))
101	100	sseq2d 3947	. . . . . . . 8 ⊢ (𝑟 = (𝑢 ∖ {∅}) → (𝐷 ⊆ ∪ 𝑟 ↔ 𝐷 ⊆ ∪ (𝑢 ∖ {∅})))
102		raleq 3358	. . . . . . . 8 ⊢ (𝑟 = (𝑢 ∖ {∅}) → (∀𝑤 ∈ 𝑟 𝑤 ≠ ∅ ↔ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅))
103	101, 102	anbi12d 633	. . . . . . 7 ⊢ (𝑟 = (𝑢 ∖ {∅}) → ((𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅) ↔ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ∧ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)))
104	103	rspcev 3571	. . . . . 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 835	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
106	105	rexlimdv3a 3245	. . . 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 1915	. . . . . 6 ⊢ Ⅎℎ𝜑
109		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎℎℝ⁺
110		nfre1 3265	. . . . . . . . . . . 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 3189	. . . . . . . . . . 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 2955	. . . . . . . . . . 11 ⊢ Ⅎℎ𝐽
113	111, 112	nfrabw 3338	. . . . . . . . . 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 2953	. . . . . . . . 9 ⊢ Ⅎℎ𝑉
115	114	nfpw 4518	. . . . . . . 8 ⊢ Ⅎℎ𝒫 𝑉
116		nfcv 2955	. . . . . . . 8 ⊢ ℲℎFin
117	115, 116	nfin 4143	. . . . . . 7 ⊢ Ⅎℎ(𝒫 𝑉 ∩ Fin)
118	117	nfcri 2943	. . . . . 6 ⊢ Ⅎℎ 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
119		nfv 1915	. . . . . 6 ⊢ Ⅎℎ(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
120	108, 118, 119	nf3an 1902	. . . . 5 ⊢ Ⅎℎ(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
121		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡ℝ⁺
122		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝐴
123		nfra1 3183	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
124		nfra1 3183	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑤 (ℎ‘𝑡) < 𝑒
125		nfra1 3183	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (ℎ‘𝑡)
126	123, 124, 125	nf3an 1902	. . . . . . . . . . . . 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	nfrex 3268	. . . . . . . . . . . 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 3189	. . . . . . . . . . 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 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐽
130	128, 129	nfrabw 3338	. . . . . . . . . 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 2953	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑉
132	131	nfpw 4518	. . . . . . . 8 ⊢ Ⅎ𝑡𝒫 𝑉
133		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑡Fin
134	132, 133	nfin 4143	. . . . . . 7 ⊢ Ⅎ𝑡(𝒫 𝑉 ∩ Fin)
135	134	nfcri 2943	. . . . . 6 ⊢ Ⅎ𝑡 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
136		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑡∪ 𝑟
137	3, 136	nfss 3907	. . . . . . 7 ⊢ Ⅎ𝑡 𝐷 ⊆ ∪ 𝑟
138		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑡∀𝑤 ∈ 𝑟 𝑤 ≠ ∅
139	137, 138	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑡(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
140	2, 135, 139	nf3an 1902	. . . . 5 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
141		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑤𝜑
142	53	nfpw 4518	. . . . . . . 8 ⊢ Ⅎ𝑤𝒫 𝑉
143		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑤Fin
144	142, 143	nfin 4143	. . . . . . 7 ⊢ Ⅎ𝑤(𝒫 𝑉 ∩ Fin)
145	144	nfcri 2943	. . . . . 6 ⊢ Ⅎ𝑤 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
146		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑤 𝐷 ⊆ ∪ 𝑟
147		nfra1 3183	. . . . . . 7 ⊢ Ⅎ𝑤∀𝑤 ∈ 𝑟 𝑤 ≠ ∅
148	146, 147	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑤(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
149	141, 145, 148	nf3an 1902	. . . . 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 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝑟 ∈ (𝒫 𝑉 ∩ Fin))
152		simp3l 1198	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐷 ⊆ ∪ 𝑟)
153		stoweidlem57.19	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ ∅)
154	153	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐷 ≠ ∅)
155		stoweidlem57.20	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
156	155	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐸 ∈ ℝ⁺)
157	26	simpld 498	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
158	157	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐵 ⊆ 𝑇)
159	66	3ad2ant1 1130	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝑉 ∈ V)
160		retop 23367	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
161	6, 160	eqeltri 2886	. . . . . . . 8 ⊢ 𝐾 ∈ Top
162		cnfex 41657	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
163	60, 161, 162	sylancl 589	. . . . . . 7 ⊢ (𝜑 → (𝐽 Cn 𝐾) ∈ V)
164	11, 10	sseqtrdi 3965	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
165	163, 164	ssexd 5192	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
166	165	3ad2ant1 1130	. . . . 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 42681	. . . 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 3245	. . 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 1915	. . . . . . 7 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑚 ∈ ℕ)
171		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑖 𝑣:(1...𝑚)⟶𝑉
172		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐷 ⊆ ∪ ran 𝑣
173		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑖 𝑦:(1...𝑚)⟶𝑌
174		nfra1 3183	. . . . . . . . . 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 1900	. . . . . . . . 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 2332	. . . . . . . 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 1902	. . . . . . 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 1900	. . . . . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
180	2, 179	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
181		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑣
182		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑡(1...𝑚)
183	181, 182, 131	nff 6483	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)⟶𝑉
184		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑡∪ ran 𝑣
185	3, 184	nfss 3907	. . . . . . . 8 ⊢ Ⅎ𝑡 𝐷 ⊆ ∪ ran 𝑣
186		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑦
187	123, 122	nfrabw 3338	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
188	150, 187	nfcxfr 2953	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑌
189	186, 182, 188	nff 6483	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑦:(1...𝑚)⟶𝑌
190		nfra1 3183	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚)
191		nfra1 3183	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡)
192	190, 191	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))
193	182, 192	nfralw 3189	. . . . . . . . . 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 1900	. . . . . . . . 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 2332	. . . . . . . 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 1902	. . . . . . 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 1900	. . . . . 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 1915	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑚 ∈ ℕ)
199		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣:(1...𝑚)⟶𝑉
200		nfv 1915	. . . . . . . 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 1902	. . . . . . 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 1900	. . . . . 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 1915	. . . . . . 7 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
205		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑣
206		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑤(1...𝑚)
207	205, 206, 53	nff 6483	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)⟶𝑉
208		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑤 𝐷 ⊆ ∪ ran 𝑣
209		nfv 1915	. . . . . . . 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 1902	. . . . . . 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 1900	. . . . . 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 2798	. . . . . 6 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
213		eqid 2798	. . . . . 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 2798	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡)))
215		eqid 2798	. . . . . 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 1233	. . . . . . 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 1407	. . . . . 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 3915	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
219	6, 9, 10, 218	fcnre 41654	. . . . . . 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 1191	. . . . . 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 21634	. . . . . . . 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 1192	. . . . . 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 6470	. . . . . . . . . . . 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 219	. . . . . . . . . 10 ⊢ (𝑦:(1...𝑚)⟶𝑌 → 𝑦:(1...𝑚)⟶{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
231	230	anim1i 617	. . . . . . . . 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 1836	. . . . . . . 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 1132	. . . . . . 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 485	. . . . . 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 7448	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
236	9, 235	eqeltrid 2894	. . . . . . 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 42696	. . . . 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 416	. . . 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 1934	. . 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 3243	. 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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 Vcvv 3441 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 𝒫 cpw 4497 {csn 4525 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Fincfn 8492 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 3c3 11681 ℝ⁺crp 12377 (,)cioo 12726 ...cfz 12885 seqcseq 13364 ↾_t crest 16686 topGenctg 16703 Topctop 21498 Clsdccld 21621 Cn ccn 21829 Compccmp 21991

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-cn 21832 df-cnp 21833 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929

This theorem is referenced by: stoweidlem58 42700

W3C validator