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Theorem metnrmlem3 22472

Description: Lemma for metnrm 22473. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 5-Sep-2015.)

**Hypotheses**
Ref	Expression
metdscn.f	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
metdscn.j	⊢ 𝐽 = (MetOpen‘𝐷)
metnrmlem.1	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
metnrmlem.2	⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))
metnrmlem.3	⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))
metnrmlem.4	⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
metnrmlem.u	⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))
metnrmlem.g	⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑇 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
metnrmlem.v	⊢ 𝑉 = ∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2))

**Assertion**
Ref	Expression
metnrmlem3	⊢ (𝜑 → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧 𝑡,𝑠,𝑤,𝑥,𝑦,𝑧,𝐷 𝐽,𝑠,𝑡,𝑤,𝑦,𝑧 𝜑,𝑠,𝑡 𝐺,𝑠,𝑡 𝑇,𝑠,𝑡,𝑤,𝑥,𝑦,𝑧 𝑆,𝑠,𝑡,𝑤,𝑥,𝑦,𝑧 𝑈,𝑠,𝑤 𝑋,𝑠,𝑡,𝑤,𝑥,𝑦,𝑧 𝐹,𝑠,𝑡,𝑤,𝑧 𝑤,𝑉,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑧,𝑤) 𝑈(𝑥,𝑦,𝑧,𝑡) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦,𝑧,𝑤) 𝐽(𝑥) 𝑉(𝑥,𝑦,𝑡,𝑠)

**Proof of Theorem metnrmlem3**
Step	Hyp	Ref	Expression
1		metnrmlem.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑇 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
2		metdscn.j	. . . 4 ⊢ 𝐽 = (MetOpen‘𝐷)
3		metnrmlem.1	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
4		metnrmlem.3	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))
5		metnrmlem.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))
6		incom 3767	. . . . 5 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
7		metnrmlem.4	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
8	6, 7	syl5eq 2656	. . . 4 ⊢ (𝜑 → (𝑇 ∩ 𝑆) = ∅)
9		metnrmlem.v	. . . 4 ⊢ 𝑉 = ∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2))
10	1, 2, 3, 4, 5, 8, 9	metnrmlem2 22471	. . 3 ⊢ (𝜑 → (𝑉 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑉))
11	10	simpld 474	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐽)
12		metdscn.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
13		metnrmlem.u	. . . 4 ⊢ 𝑈 = ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))
14	12, 2, 3, 5, 4, 7, 13	metnrmlem2 22471	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈))
15	14	simpld 474	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
16	10	simprd 478	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
17	14	simprd 478	. 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
18	9	ineq1i 3772	. . . 4 ⊢ (𝑉 ∩ 𝑈) = (∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
19		iunin1 4521	. . . 4 ⊢ ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) = (∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
20	18, 19	eqtr4i 2635	. . 3 ⊢ (𝑉 ∩ 𝑈) = ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
21	13	ineq2i 3773	. . . . . . . 8 ⊢ ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) = ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)))
22		iunin2 4520	. . . . . . . 8 ⊢ ∪ 𝑡 ∈ 𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) = ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ ∪ 𝑡 ∈ 𝑇 (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)))
23	21, 22	eqtr4i 2635	. . . . . . 7 ⊢ ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) = ∪ 𝑡 ∈ 𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)))
24	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝐷 ∈ (∞Met‘𝑋))
25		eqid 2610	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
26	25	cldss 20643	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
27	5, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
28	2	mopnuni 22056	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	3, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
30	27, 29	sseqtr4d 3605	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
31	30	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑋)
32	31	adantrr 749	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑠 ∈ 𝑋)
33	25	cldss 20643	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
34	4, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
35	34, 29	sseqtr4d 3605	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
36	35	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑋)
37	36	adantrl 748	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑡 ∈ 𝑋)
38	1, 2, 3, 4, 5, 8	metnrmlem1a 22469	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0 < (𝐺‘𝑠) ∧ if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺))
39	38	simprd 478	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
40	39	adantrr 749	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
41	40	rphalfcld 11760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ⁺)
42	41	rpxrd 11749	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ^*)
43	12, 2, 3, 5, 4, 7	metnrmlem1a 22469	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
44	43	adantrl 748	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
45	44	simprd 478	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺)
46	45	rphalfcld 11760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ⁺)
47	46	rpxrd 11749	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ^*)
48	40	rpred 11748	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ)
49	48	rehalfcld 11156	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ)
50	45	rpred 11748	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ)
51	50	rehalfcld 11156	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ)
52		rexadd 11937	. . . . . . . . . . . . . . 15 ⊢ (((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ ∧ (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) +_𝑒 (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) = ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) + (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)))
53	49, 51, 52	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) +_𝑒 (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) = ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) + (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)))
54	48	recnd 9947	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℂ)
55	50	recnd 9947	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℂ)
56		2cnd 10970	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℂ)
57		2ne0 10990	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
58	57	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ≠ 0)
59	54, 55, 56, 58	divdird 10718	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) = ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) + (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)))
60	53, 59	eqtr4d 2647	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) +_𝑒 (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) = ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2))
61	1, 2, 3, 4, 5, 8	metnrmlem1 22470	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑠 ∈ 𝑆)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
62	61	ancom2s 840	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
63		xmetsym 21962	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
64	24, 37, 32, 63	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
65	62, 64	breqtrd 4609	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡))
66	12, 2, 3, 5, 4, 7	metnrmlem1 22470	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡))
67	40	rpxrd 11749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ^*)
68	45	rpxrd 11749	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ^*)
69		xmetcl 21946	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋) → (𝑠𝐷𝑡) ∈ ℝ^*)
70	24, 32, 37, 69	syl3anc 1318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑠𝐷𝑡) ∈ ℝ^*)
71		xle2add 11961	. . . . . . . . . . . . . . . . 17 ⊢ (((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ^* ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ^) ∧ ((𝑠𝐷𝑡) ∈ ℝ^ ∧ (𝑠𝐷𝑡) ∈ ℝ^*)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ≤ ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡))))
72	67, 68, 70, 70, 71	syl22anc 1319	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ≤ ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡))))
73	65, 66, 72	mp2and 711	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ≤ ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡)))
74	48, 50	readdcld 9948	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℝ)
75	74	recnd 9947	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℂ)
76	75, 56, 58	divcan2d 10682	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (2 · ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) = (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))))
77		2re 10967	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
78	74	rehalfcld 11156	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ)
79		rexmul 11973	. . . . . . . . . . . . . . . . 17 ⊢ ((2 ∈ ℝ ∧ ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ) → (2 ·_e ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) = (2 · ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)))
80	77, 78, 79	sylancr 694	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (2 ·_e ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) = (2 · ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)))
81		rexadd 11937	. . . . . . . . . . . . . . . . 17 ⊢ ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) = (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))))
82	48, 50, 81	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) = (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))))
83	76, 80, 82	3eqtr4d 2654	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (2 ·_e ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) = (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))))
84		x2times 12001	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠𝐷𝑡) ∈ ℝ^* → (2 ·_e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡)))
85	70, 84	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (2 ·_e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡)))
86	73, 83, 85	3brtr4d 4615	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (2 ·_e ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) ≤ (2 ·_e (𝑠𝐷𝑡)))
87	78	rexrd 9968	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ^*)
88		2rp 11713	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
89	88	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℝ⁺)
90		xlemul2 11993	. . . . . . . . . . . . . . 15 ⊢ ((((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ^* ∧ (𝑠𝐷𝑡) ∈ ℝ^* ∧ 2 ∈ ℝ⁺) → (((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ≤ (𝑠𝐷𝑡) ↔ (2 ·_e ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) ≤ (2 ·_e (𝑠𝐷𝑡))))
91	87, 70, 89, 90	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ≤ (𝑠𝐷𝑡) ↔ (2 ·_e ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2)) ≤ (2 ·_e (𝑠𝐷𝑡))))
92	86, 91	mpbird 246	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ≤ (𝑠𝐷𝑡))
93	60, 92	eqbrtrd 4605	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) +_𝑒 (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ≤ (𝑠𝐷𝑡))
94		bldisj 22013	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋) ∧ ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ^* ∧ (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ^* ∧ ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) +_𝑒 (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ≤ (𝑠𝐷𝑡))) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) = ∅)
95	24, 32, 37, 42, 47, 93, 94	syl33anc 1333	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) = ∅)
96		eqimss 3620	. . . . . . . . . . 11 ⊢ (((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) = ∅ → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅)
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅)
98	97	anassrs 678	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝑆) ∧ 𝑡 ∈ 𝑇) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅)
99	98	ralrimiva 2949	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ∀𝑡 ∈ 𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅)
100		iunss 4497	. . . . . . . 8 ⊢ (∪ 𝑡 ∈ 𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅ ↔ ∀𝑡 ∈ 𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅)
101	99, 100	sylibr 223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ∪ 𝑡 ∈ 𝑇 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ (𝑡(ball‘𝐷)(if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2))) ⊆ ∅)
102	23, 101	syl5eqss 3612	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
103	102	ralrimiva 2949	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
104		iunss 4497	. . . . 5 ⊢ (∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅ ↔ ∀𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
105	103, 104	sylibr 223	. . . 4 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
106		ss0 3926	. . . 4 ⊢ (∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅ → ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) = ∅)
107	105, 106	syl 17	. . 3 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) = ∅)
108	20, 107	syl5eq 2656	. 2 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = ∅)
109		sseq2 3590	. . . 4 ⊢ (𝑧 = 𝑉 → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ 𝑉))
110		ineq1 3769	. . . . 5 ⊢ (𝑧 = 𝑉 → (𝑧 ∩ 𝑤) = (𝑉 ∩ 𝑤))
111	110	eqeq1d 2612	. . . 4 ⊢ (𝑧 = 𝑉 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑤) = ∅))
112	109, 111	3anbi13d 1393	. . 3 ⊢ (𝑧 = 𝑉 → ((𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅)))
113		sseq2 3590	. . . 4 ⊢ (𝑤 = 𝑈 → (𝑇 ⊆ 𝑤 ↔ 𝑇 ⊆ 𝑈))
114		ineq2 3770	. . . . 5 ⊢ (𝑤 = 𝑈 → (𝑉 ∩ 𝑤) = (𝑉 ∩ 𝑈))
115	114	eqeq1d 2612	. . . 4 ⊢ (𝑤 = 𝑈 → ((𝑉 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑈) = ∅))
116	113, 115	3anbi23d 1394	. . 3 ⊢ (𝑤 = 𝑈 → ((𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈 ∧ (𝑉 ∩ 𝑈) = ∅)))
117	112, 116	rspc2ev 3295	. 2 ⊢ ((𝑉 ∈ 𝐽 ∧ 𝑈 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈 ∧ (𝑉 ∩ 𝑈) = ∅)) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
118	11, 15, 16, 17, 108, 117	syl113anc 1330	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ∪ cuni 4372 ∪ ciun 4455 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 infcinf 8230 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 / cdiv 10563 2c2 10947 ℝ⁺crp 11708 +_𝑒 cxad 11820 ·_e cxmu 11821 ∞Metcxmt 19552 ballcbl 19554 MetOpencmopn 19557 Clsdccld 20630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-ec 7631 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-bl 19562 df-mopn 19563 df-top 20521 df-bases 20522 df-topon 20523 df-cld 20633 df-ntr 20634 df-cls 20635

This theorem is referenced by: metnrm 22473

W3C validator