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Theorem metnrmlem3 23469

Description: Lemma for metnrm 23470. (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 4178	. . . . 5 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
7		metnrmlem.4	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
8	6, 7	syl5eq 2868	. . . 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 23468	. . 3 ⊢ (𝜑 → (𝑉 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑉))
11	10	simpld 497	. 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 23468	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈))
15	14	simpld 497	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
16	10	simprd 498	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
17	14	simprd 498	. 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
18	9	ineq1i 4185	. . . 4 ⊢ (𝑉 ∩ 𝑈) = (∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
19		iunin1 4994	. . . 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 2847	. . 3 ⊢ (𝑉 ∩ 𝑈) = ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
21	13	ineq2i 4186	. . . . . . . 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 4993	. . . . . . . 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 2847	. . . . . . 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 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝐷 ∈ (∞Met‘𝑋))
25		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
26	25	cldss 21637	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
27	5, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
28	2	mopnuni 23051	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	3, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
30	27, 29	sseqtrrd 4008	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
31	30	sselda 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑋)
32	31	adantrr 715	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑠 ∈ 𝑋)
33	25	cldss 21637	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
34	4, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
35	34, 29	sseqtrrd 4008	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
36	35	sselda 3967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑋)
37	36	adantrl 714	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑡 ∈ 𝑋)
38	1, 2, 3, 4, 5, 8	metnrmlem1a 23466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0 < (𝐺‘𝑠) ∧ if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺))
39	38	simprd 498	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
40	39	adantrr 715	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
41	40	rphalfcld 12444	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ⁺)
42	41	rpxrd 12433	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ^*)
43	12, 2, 3, 5, 4, 7	metnrmlem1a 23466	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
44	43	adantrl 714	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
45	44	simprd 498	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺)
46	45	rphalfcld 12444	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ⁺)
47	46	rpxrd 12433	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ^*)
48	40	rpred 12432	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ)
49	48	rehalfcld 11885	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ)
50	45	rpred 12432	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ)
51	50	rehalfcld 11885	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ)
52	49, 51	rexaddd 12628	. . . . . . . . . . . . . 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)))
53	48	recnd 10669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℂ)
54	50	recnd 10669	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℂ)
55		2cnd 11716	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℂ)
56		2ne0 11742	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
57	56	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ≠ 0)
58	53, 54, 55, 57	divdird 11454	. . . . . . . . . . . . . 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)))
59	52, 58	eqtr4d 2859	. . . . . . . . . . . . 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))
60	1, 2, 3, 4, 5, 8	metnrmlem1 23467	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑠 ∈ 𝑆)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
61	60	ancom2s 648	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
62		xmetsym 22957	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
63	24, 37, 32, 62	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
64	61, 63	breqtrd 5092	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡))
65	12, 2, 3, 5, 4, 7	metnrmlem1 23467	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡))
66	40	rpxrd 12433	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ^*)
67	45	rpxrd 12433	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ^*)
68		xmetcl 22941	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋) → (𝑠𝐷𝑡) ∈ ℝ^*)
69	24, 32, 37, 68	syl3anc 1367	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑠𝐷𝑡) ∈ ℝ^*)
70		xle2add 12653	. . . . . . . . . . . . . . . . 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))))
71	66, 67, 69, 69, 70	syl22anc 836	. . . . . . . . . . . . . . . 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))))
72	64, 65, 71	mp2and 697	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) +_𝑒 if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ≤ ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡)))
73	48, 50	readdcld 10670	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℝ)
74	73	recnd 10669	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℂ)
75	74, 55, 57	divcan2d 11418	. . . . . . . . . . . . . . . 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))))
76		2re 11712	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
77	73	rehalfcld 11885	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ)
78		rexmul 12665	. . . . . . . . . . . . . . . . 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)))
79	76, 77, 78	sylancr 589	. . . . . . . . . . . . . . . 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)))
80	48, 50	rexaddd 12628	. . . . . . . . . . . . . . . 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))))
81	75, 79, 80	3eqtr4d 2866	. . . . . . . . . . . . . . 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))))
82		x2times 12693	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠𝐷𝑡) ∈ ℝ^* → (2 ·_e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡)))
83	69, 82	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (2 ·_e (𝑠𝐷𝑡)) = ((𝑠𝐷𝑡) +_𝑒 (𝑠𝐷𝑡)))
84	72, 81, 83	3brtr4d 5098	. . . . . . . . . . . . . 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)))
85	77	rexrd 10691	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ^*)
86		2rp 12395	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
87	86	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℝ⁺)
88		xlemul2 12685	. . . . . . . . . . . . . . 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))))
89	85, 69, 87, 88	syl3anc 1367	. . . . . . . . . . . . . 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))))
90	84, 89	mpbird 259	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ≤ (𝑠𝐷𝑡))
91	59, 90	eqbrtrd 5088	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) +_𝑒 (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2)) ≤ (𝑠𝐷𝑡))
92		bldisj 23008	. . . . . . . . . . . 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))) = ∅)
93	24, 32, 37, 42, 47, 91, 92	syl33anc 1381	. . . . . . . . . . 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))) = ∅)
94		eqimss 4023	. . . . . . . . . . 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))) ⊆ ∅)
95	93, 94	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))) ⊆ ∅)
96	95	anassrs 470	. . . . . . . . 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))) ⊆ ∅)
97	96	ralrimiva 3182	. . . . . . . 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))) ⊆ ∅)
98		iunss 4969	. . . . . . . 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))) ⊆ ∅)
99	97, 98	sylibr 236	. . . . . . 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))) ⊆ ∅)
100	23, 99	eqsstrid 4015	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
101	100	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
102		iunss 4969	. . . . 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) ⊆ ∅)
103	101, 102	sylibr 236	. . . 4 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
104		ss0 4352	. . . 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) = ∅)
105	103, 104	syl 17	. . 3 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) = ∅)
106	20, 105	syl5eq 2868	. 2 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = ∅)
107		sseq2 3993	. . . 4 ⊢ (𝑧 = 𝑉 → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ 𝑉))
108		ineq1 4181	. . . . 5 ⊢ (𝑧 = 𝑉 → (𝑧 ∩ 𝑤) = (𝑉 ∩ 𝑤))
109	108	eqeq1d 2823	. . . 4 ⊢ (𝑧 = 𝑉 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑤) = ∅))
110	107, 109	3anbi13d 1434	. . 3 ⊢ (𝑧 = 𝑉 → ((𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅)))
111		sseq2 3993	. . . 4 ⊢ (𝑤 = 𝑈 → (𝑇 ⊆ 𝑤 ↔ 𝑇 ⊆ 𝑈))
112		ineq2 4183	. . . . 5 ⊢ (𝑤 = 𝑈 → (𝑉 ∩ 𝑤) = (𝑉 ∩ 𝑈))
113	112	eqeq1d 2823	. . . 4 ⊢ (𝑤 = 𝑈 → ((𝑉 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑈) = ∅))
114	111, 113	3anbi23d 1435	. . 3 ⊢ (𝑤 = 𝑈 → ((𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈 ∧ (𝑉 ∩ 𝑈) = ∅)))
115	110, 114	rspc2ev 3635	. 2 ⊢ ((𝑉 ∈ 𝐽 ∧ 𝑈 ∈ 𝐽 ∧ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈 ∧ (𝑉 ∩ 𝑈) = ∅)) → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))
116	11, 15, 16, 17, 106, 115	syl113anc 1378	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ∪ cuni 4838 ∪ ciun 4919 class class class wbr 5066 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 (class class class)co 7156 infcinf 8905 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 / cdiv 11297 2c2 11693 ℝ⁺crp 12390 +_𝑒 cxad 12506 ·_e cxmu 12507 ∞Metcxmet 20530 ballcbl 20532 MetOpencmopn 20535 Clsdccld 21624

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-ec 8291 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-bl 20540 df-mopn 20541 df-top 21502 df-topon 21519 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629

This theorem is referenced by: metnrm 23470

W3C validator