Theorem stoweidlem57 47270

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:   𝐴,𝑎,𝑓,𝑞,𝑟   𝑇,𝑟,𝑤   𝑈,𝑔   𝑔,𝑉   𝑡,𝐾   𝜑,𝑟   𝑔,ℎ,𝜑   𝑓,𝑉,𝑟   𝑇,𝑎,𝑒   𝑇,ℎ,𝑥   𝑤,𝑌   𝑔,𝐽,ℎ,𝑡   𝑤,𝐽   𝑈,𝑒,ℎ,𝑤   𝑒,𝐸,𝑤   𝑓,𝐸,𝑔,𝑡,𝑥   𝑈,𝑎,𝑓,𝑞,𝑟   𝜑,𝑎,𝑓,𝑞   𝐵,𝑓,𝑔,𝑥   𝐷,𝑎,𝑓,𝑞,𝑟   𝑤,𝐷,ℎ   𝑥,𝐷   𝑓,ℎ   𝜑,𝑒,𝑤   ℎ,𝐸,𝑟,𝑡,𝑤   𝐵,𝑟,𝑤   𝑇,𝑓,𝑔,𝑞,𝑡   𝐴,𝑒,𝑡,𝑎   𝐷,𝑒,𝑔,𝑓   𝐴,𝑔,ℎ,𝑥   𝑓,𝑌,𝑔,𝑟   𝑤,𝐴

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 𝑖 𝑦 𝑘 𝑢 𝑚 𝑣 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 2918	. . . . . . . . . . 11 ⊢ Ⅎ𝑡 𝑠 ∈ 𝐷
5	2, 4	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑠 ∈ 𝐷)
6		stoweidlem57.6	. . . . . . . . . 10 ⊢ 𝐾 = (topGen‘ran (,))
7		stoweidlem57.10	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Comp)
8	7	adantr 486	. . . . . . . . . 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 486	. . . . . . . . . 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 1196	. . . . . . . . . 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 1196	. . . . . . . . . 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 728	. . . . . . . . . 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 728	. . . . . . . . . 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 23740	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
24	9	iscld 23372	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → (𝐵 ∈ (Clsd‘𝐽) ↔ (𝐵 ⊆ 𝑇 ∧ (𝑇 ∖ 𝐵) ∈ 𝐽)))
25	7, 23, 24	3syl 19	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑇 ∖ 𝐵) ∈ 𝐽)
28	21, 27	eqeltrid 2868	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
29	28	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑈 ∈ 𝐽)
30		stoweidlem57.17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ (Clsd‘𝐽))
31	9	cldss 23374	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (Clsd‘𝐽) → 𝐷 ⊆ 𝑇)
32	30, 31	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
33	32	sselda 3913	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ 𝑇)
34		stoweidlem57.18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)
35		disjr 4407	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∩ 𝐷) = ∅ ↔ ∀𝑠 ∈ 𝐷 ¬ 𝑠 ∈ 𝐵)
36	34, 35	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑠 ∈ 𝐷 ¬ 𝑠 ∈ 𝐵)
37	36	r19.21bi 3258	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → ¬ 𝑠 ∈ 𝐵)
38	33, 37	eldifd 4053	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ (𝑇 ∖ 𝐵))
39	38, 21	eleqtrrdi 2875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ 𝑈)
40	1, 5, 6, 8, 9, 10, 12, 14, 16, 18, 20, 29, 39	stoweidlem56 47269	. . . . . . . . 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 488	. . . . . . . . . . 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 791	. . . . . . . . . . 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 785	. . . . . . . . . . . 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 3438	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 525	. . . . . . . . . 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 3099	. . . . . . . . 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 3096	. . . . . . . . 9 ⊢ (∃𝑤 ∈ 𝐽 (𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
50	40, 48, 49	3syl 19	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
51		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑠
52		nfrab1 3435	. . . . . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑉
54	51, 53	elunif 46236	. . . . . . . 8 ⊢ (𝑠 ∈ ∪ 𝑉 ↔ ∃𝑤(𝑠 ∈ 𝑤 ∧ 𝑤 ∈ 𝑉))
55	50, 54	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝐷) → 𝑠 ∈ ∪ 𝑉)
56	55	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ 𝐷 → 𝑠 ∈ ∪ 𝑉))
57	56	ssrdv 3919	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑉)
58		cmpcld 23747	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ 𝐷 ∈ (Clsd‘𝐽)) → (𝐽 ↾t 𝐷) ∈ Comp)
59	7, 30, 58	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t 𝐷) ∈ Comp)
60	7, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Top)
61	9	cmpsub 23745	. . . . . . . 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 596	. . . . . . 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 4008	. . . . . . . 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 3959	. . . . . . 7 ⊢ 𝑉 ⊆ 𝐽
66	44, 7	rabexd 5305	. . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ V)
67		elpwg 4563	. . . . . . . 8 ⊢ (𝑉 ∈ V → (𝑉 ∈ 𝒫 𝐽 ↔ 𝑉 ⊆ 𝐽))
68	66, 67	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝒫 𝐽 ↔ 𝑉 ⊆ 𝐽))
69	65, 68	mpbiri 261	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝒫 𝐽)
70		unieq 4881	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → ∪ 𝑘 = ∪ 𝑉)
71	70	sseq2d 3945	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (𝐷 ⊆ ∪ 𝑘 ↔ 𝐷 ⊆ ∪ 𝑉))
72		pweq 4574	. . . . . . . . . 10 ⊢ (𝑘 = 𝑉 → 𝒫 𝑘 = 𝒫 𝑉)
73	72	ineq1d 4119	. . . . . . . . 9 ⊢ (𝑘 = 𝑉 → (𝒫 𝑘 ∩ Fin) = (𝒫 𝑉 ∩ Fin))
74	73	rexeqdv 3324	. . . . . . . 8 ⊢ (𝑘 = 𝑉 → (∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢 ↔ ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
75	71, 74	imbi12d 347	. . . . . . 7 ⊢ (𝑘 = 𝑉 → ((𝐷 ⊆ ∪ 𝑘 → ∃𝑢 ∈ (𝒫 𝑘 ∩ Fin)𝐷 ⊆ ∪ 𝑢) ↔ (𝐷 ⊆ ∪ 𝑉 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢)))
76	75	rspccva 3579	. . . . . 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 596	. . . . 5 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑉 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢))
78	57, 77	mpd 16	. . . 4 ⊢ (𝜑 → ∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢)
79		elinel1 4101	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → 𝑢 ∈ 𝒫 𝑉)
80		elpwi 4567	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝒫 𝑉 → 𝑢 ⊆ 𝑉)
81	80	ssdifssd 4086	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 𝑉 → (𝑢 ∖ {∅}) ⊆ 𝑉)
82		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
83		difexg 5295	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ V → (𝑢 ∖ {∅}) ∈ V)
84	82, 83	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑢 ∖ {∅}) ∈ V
85	84	elpw 4564	. . . . . . . . . 10 ⊢ ((𝑢 ∖ {∅}) ∈ 𝒫 𝑉 ↔ (𝑢 ∖ {∅}) ⊆ 𝑉)
86	81, 85	sylibr 237	. . . . . . . . 9 ⊢ (𝑢 ∈ 𝒫 𝑉 → (𝑢 ∖ {∅}) ∈ 𝒫 𝑉)
87	79, 86	syl 18	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ 𝒫 𝑉)
88		elinel2 4102	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → 𝑢 ∈ Fin)
89		diffi 9215	. . . . . . . . 9 ⊢ (𝑢 ∈ Fin → (𝑢 ∖ {∅}) ∈ Fin)
90	88, 89	syl 18	. . . . . . . 8 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ Fin)
91	87, 90	elind 4100	. . . . . . 7 ⊢ (𝑢 ∈ (𝒫 𝑉 ∩ Fin) → (𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin))
92	91	3ad2ant2 1152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → (𝑢 ∖ {∅}) ∈ (𝒫 𝑉 ∩ Fin))
93		unidif0 5325	. . . . . . . . 9 ⊢ ∪ (𝑢 ∖ {∅}) = ∪ 𝑢
94	93	sseq2i 3942	. . . . . . . 8 ⊢ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ↔ 𝐷 ⊆ ∪ 𝑢)
95	94	biimpri 231	. . . . . . 7 ⊢ (𝐷 ⊆ ∪ 𝑢 → 𝐷 ⊆ ∪ (𝑢 ∖ {∅}))
96	95	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → 𝐷 ⊆ ∪ (𝑢 ∖ {∅}))
97		eldifsni 4753	. . . . . . . 8 ⊢ (𝑤 ∈ (𝑢 ∖ {∅}) → 𝑤 ≠ ∅)
98	97	rgen 3082	. . . . . . 7 ⊢ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅
99	98	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)
100		unieq 4881	. . . . . . . . 9 ⊢ (𝑟 = (𝑢 ∖ {∅}) → ∪ 𝑟 = ∪ (𝑢 ∖ {∅}))
101	100	sseq2d 3945	. . . . . . . 8 ⊢ (𝑟 = (𝑢 ∖ {∅}) → (𝐷 ⊆ ∪ 𝑟 ↔ 𝐷 ⊆ ∪ (𝑢 ∖ {∅})))
102		raleq 3320	. . . . . . . 8 ⊢ (𝑟 = (𝑢 ∖ {∅}) → (∀𝑤 ∈ 𝑟 𝑤 ≠ ∅ ↔ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅))
103	101, 102	anbi12d 644	. . . . . . 7 ⊢ (𝑟 = (𝑢 ∖ {∅}) → ((𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅) ↔ (𝐷 ⊆ ∪ (𝑢 ∖ {∅}) ∧ ∀𝑤 ∈ (𝑢 ∖ {∅})𝑤 ≠ ∅)))
104	103	rspcev 3580	. . . . . 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 850	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝒫 𝑉 ∩ Fin) ∧ 𝐷 ⊆ ∪ 𝑢) → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
106	105	rexlimdv3a 3171	. . . 4 ⊢ (𝜑 → (∃𝑢 ∈ (𝒫 𝑉 ∩ Fin)𝐷 ⊆ ∪ 𝑢 → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)))
107	78, 106	mpd 16	. . 3 ⊢ (𝜑 → ∃𝑟 ∈ (𝒫 𝑉 ∩ Fin)(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
108		nfv 1949	. . . . . 6 ⊢ Ⅎℎ𝜑
109		nfcv 2926	. . . . . . . . . . . 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 3313	. . . . . . . . . . 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 2926	. . . . . . . . . . 11 ⊢ Ⅎℎ𝐽
113	111, 112	nfrabw 3451	. . . . . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎℎ𝑉
115	114	nfpw 4580	. . . . . . . 8 ⊢ Ⅎℎ𝒫 𝑉
116		nfcv 2926	. . . . . . . 8 ⊢ ℲℎFin
117	115, 116	nfin 4124	. . . . . . 7 ⊢ Ⅎℎ(𝒫 𝑉 ∩ Fin)
118	117	nfcri 2918	. . . . . 6 ⊢ Ⅎℎ 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
119		nfv 1949	. . . . . 6 ⊢ Ⅎℎ(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
120	108, 118, 119	nf3an 1936	. . . . 5 ⊢ Ⅎℎ(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
121		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡ℝ+
122		nfcv 2926	. . . . . . . . . . . . 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 1936	. . . . . . . . . . . . 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 3314	. . . . . . . . . . . 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 3313	. . . . . . . . . . 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 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝐽
130	128, 129	nfrabw 3451	. . . . . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑉
132	131	nfpw 4580	. . . . . . . 8 ⊢ Ⅎ𝑡𝒫 𝑉
133		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡Fin
134	132, 133	nfin 4124	. . . . . . 7 ⊢ Ⅎ𝑡(𝒫 𝑉 ∩ Fin)
135	134	nfcri 2918	. . . . . 6 ⊢ Ⅎ𝑡 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
136		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡∪ 𝑟
137	3, 136	nfss 3906	. . . . . . 7 ⊢ Ⅎ𝑡 𝐷 ⊆ ∪ 𝑟
138		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑡∀𝑤 ∈ 𝑟 𝑤 ≠ ∅
139	137, 138	nfan 1934	. . . . . 6 ⊢ Ⅎ𝑡(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
140	2, 135, 139	nf3an 1936	. . . . 5 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅))
141		nfv 1949	. . . . . 6 ⊢ Ⅎ𝑤𝜑
142	53	nfpw 4580	. . . . . . . 8 ⊢ Ⅎ𝑤𝒫 𝑉
143		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑤Fin
144	142, 143	nfin 4124	. . . . . . 7 ⊢ Ⅎ𝑤(𝒫 𝑉 ∩ Fin)
145	144	nfcri 2918	. . . . . 6 ⊢ Ⅎ𝑤 𝑟 ∈ (𝒫 𝑉 ∩ Fin)
146		nfv 1949	. . . . . . 7 ⊢ Ⅎ𝑤 𝐷 ⊆ ∪ 𝑟
147		nfra1 3290	. . . . . . 7 ⊢ Ⅎ𝑤∀𝑤 ∈ 𝑟 𝑤 ≠ ∅
148	146, 147	nfan 1934	. . . . . 6 ⊢ Ⅎ𝑤(𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)
149	141, 145, 148	nf3an 1936	. . . . 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 1155	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝑟 ∈ (𝒫 𝑉 ∩ Fin))
152		simp3l 1220	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐷 ⊆ ∪ 𝑟)
153		stoweidlem57.19	. . . . . 6 ⊢ (𝜑 → 𝐷 ≠ ∅)
154	153	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐷 ≠ ∅)
155		stoweidlem57.20	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
156	155	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐸 ∈ ℝ+)
157	26	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
158	157	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝐵 ⊆ 𝑇)
159	66	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ (𝒫 𝑉 ∩ Fin) ∧ (𝐷 ⊆ ∪ 𝑟 ∧ ∀𝑤 ∈ 𝑟 𝑤 ≠ ∅)) → 𝑉 ∈ V)
160		retop 25107	. . . . . . . . 9 ⊢ (topGen‘ran (,)) ∈ Top
161	6, 160	eqeltri 2860	. . . . . . . 8 ⊢ 𝐾 ∈ Top
162		cnfex 46248	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
163	60, 161, 162	sylancl 598	. . . . . . 7 ⊢ (𝜑 → (𝐽 Cn 𝐾) ∈ V)
164	11, 10	sseqtrdi 3953	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
165	163, 164	ssexd 5290	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
166	165	3ad2ant1 1151	. . . . 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 47252	. . . 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 3171	. . 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 16	. 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 1949	. . . . . . 7 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑚 ∈ ℕ)
171		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑖 𝑣:(1...𝑚)⟶𝑉
172		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑖 𝐷 ⊆ ∪ ran 𝑣
173		nfv 1949	. . . . . . . . . 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 1934	. . . . . . . . 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 2358	. . . . . . . 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 1936	. . . . . . 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 1934	. . . . . 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 1949	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑚 ∈ ℕ
180	2, 179	nfan 1934	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑚 ∈ ℕ)
181		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑣
182		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑡(1...𝑚)
183	181, 182, 131	nff 6724	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑣:(1...𝑚)⟶𝑉
184		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑡∪ ran 𝑣
185	3, 184	nfss 3906	. . . . . . . 8 ⊢ Ⅎ𝑡 𝐷 ⊆ ∪ ran 𝑣
186		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑦
187	123, 122	nfrabw 3451	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
188	150, 187	nfcxfr 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝑌
189	186, 182, 188	nff 6724	. . . . . . . . . 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 1934	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(∀𝑡 ∈ (𝑣‘𝑖)((𝑦‘𝑖)‘𝑡) < (𝐸 / 𝑚) ∧ ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑚)) < ((𝑦‘𝑖)‘𝑡))
193	182, 192	nfralw 3313	. . . . . . . . . 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 1934	. . . . . . . . 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 2358	. . . . . . . 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 1936	. . . . . . 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 1934	. . . . . 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 1949	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑚 ∈ ℕ)
199		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑣:(1...𝑚)⟶𝑉
200		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐷 ⊆ ∪ ran 𝑣
201		nfe1 2190	. . . . . . . 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 1936	. . . . . . 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 1934	. . . . . 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 1949	. . . . . . 7 ⊢ Ⅎ𝑤(𝜑 ∧ 𝑚 ∈ ℕ)
205		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑤𝑣
206		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑤(1...𝑚)
207	205, 206, 53	nff 6724	. . . . . . . 8 ⊢ Ⅎ𝑤 𝑣:(1...𝑚)⟶𝑉
208		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑤 𝐷 ⊆ ∪ ran 𝑣
209		nfv 1949	. . . . . . . 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 1936	. . . . . . 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 1934	. . . . . 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 2763	. . . . . 6 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
213		eqid 2763	. . . . . 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 2763	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑚) ↦ ((𝑦‘𝑖)‘𝑡)))
215		eqid 2763	. . . . . 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 1255	. . . . . . 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 1438	. . . . . 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 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
219	6, 9, 10, 218	fcnre 46245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
220	219	ad4ant14 765	. . . . . 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 781	. . . . . 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 1213	. . . . . 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 23374	. . . . . . . 8 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ 𝑇)
224	22, 223	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
225	224	ad2antrr 739	. . . . . 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 1214	. . . . . 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 739	. . . . . 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 6708	. . . . . . . . . . . 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 627	. . . . . . . . 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 1868	. . . . . . . 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 1153	. . . . . . 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 487	. . . . . 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 7780	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝐽 ∈ V)
236	9, 235	eqeltrid 2868	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ V)
237	236	ad2antrr 739	. . . . . 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 739	. . . . . 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 739	. . . . . 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 47267	. . . . 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 418	. . . 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 1968	. . 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 3167	. 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 16	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:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189  ran crn 5656  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∈ cmpo 7442  Fincfn 8998  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   · cmul 11229   < clt 11367   ≤ cle 11368   − cmin 11565   / cdiv 11997  ℕcn 12359  3c3 12422  ℝ⁺crp 13144  (,)cioo 13500  ...cfz 13663  seqcseq 14168   ↾_t crest 17617  topGenctg 17634  Topctop 23238  Clsdccld 23361   Cn ccn 23569  Compccmp 23731

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-inf2 9666  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-isom 6564  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-er 8742  df-map 8874  df-pm 8875  df-ixp 8951  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-fi 9427  df-sup 9458  df-inf 9459  df-oi 9528  df-card 10044  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-q 13100  df-rp 13145  df-xneg 13265  df-xadd 13266  df-xmul 13267  df-ioo 13504  df-ico 13506  df-icc 13507  df-fz 13664  df-fzo 13813  df-fl 13956  df-seq 14169  df-exp 14229  df-hash 14499  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-rlim 15682  df-sum 15880  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-starv 17469  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-unif 17477  df-hom 17478  df-cco 17479  df-rest 17619  df-topn 17620  df-0g 17638  df-gsum 17639  df-topgen 17640  df-pt 17641  df-prds 17644  df-xrs 17700  df-qtop 17705  df-imas 17706  df-xps 17708  df-mre 17782  df-mrc 17783  df-acs 17785  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-submnd 19005  df-mulg 19304  df-cntz 19558  df-cmn 20023  df-psmet 21697  df-xmet 21698  df-met 21699  df-bl 21700  df-mopn 21701  df-cnfld 21706  df-top 23239  df-topon 23256  df-topsp 23278  df-bases 23291  df-cld 23364  df-cn 23572  df-cnp 23573  df-cmp 23732  df-tx 23908  df-hmeo 24101  df-xms 24666  df-ms 24667  df-tms 24668

This theorem is used by:  stoweidlem58  47271