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Theorem metnrmlem3 24897

Description: Lemma for metnrm 24898. (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 4217	. . . . 5 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
7		metnrmlem.4	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
8	6, 7	eqtrid 2787	. . . 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 24896	. . 3 ⊢ (𝜑 → (𝑉 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑉))
11	10	simpld 494	. 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 24896	. . 3 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ∧ 𝑇 ⊆ 𝑈))
15	14	simpld 494	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
16	10	simprd 495	. 2 ⊢ (𝜑 → 𝑆 ⊆ 𝑉)
17	14	simprd 495	. 2 ⊢ (𝜑 → 𝑇 ⊆ 𝑈)
18	9	ineq1i 4224	. . . 4 ⊢ (𝑉 ∩ 𝑈) = (∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
19		iunin1 5077	. . . 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 2766	. . 3 ⊢ (𝑉 ∩ 𝑈) = ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
21	13	ineq2i 4225	. . . . . . . 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 5076	. . . . . . . 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 2766	. . . . . . 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 2735	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
26	25	cldss 23053	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
27	5, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
28	2	mopnuni 24467	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	3, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
30	27, 29	sseqtrrd 4037	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
31	30	sselda 3995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑋)
32	31	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑠 ∈ 𝑋)
33	25	cldss 23053	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
34	4, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
35	34, 29	sseqtrrd 4037	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
36	35	sselda 3995	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑋)
37	36	adantrl 716	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑡 ∈ 𝑋)
38	1, 2, 3, 4, 5, 8	metnrmlem1a 24894	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (0 < (𝐺‘𝑠) ∧ if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺))
39	38	simprd 495	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
40	39	adantrr 717	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
41	40	rphalfcld 13087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ⁺)
42	41	rpxrd 13076	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ^*)
43	12, 2, 3, 5, 4, 7	metnrmlem1a 24894	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
44	43	adantrl 716	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
45	44	simprd 495	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺)
46	45	rphalfcld 13087	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ⁺)
47	46	rpxrd 13076	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ^*)
48	40	rpred 13075	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ)
49	48	rehalfcld 12511	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ)
50	45	rpred 13075	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ)
51	50	rehalfcld 12511	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ)
52	49, 51	rexaddd 13273	. . . . . . . . . . . . . 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 11287	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℂ)
54	50	recnd 11287	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℂ)
55		2cnd 12342	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℂ)
56		2ne0 12368	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
57	56	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ≠ 0)
58	53, 54, 55, 57	divdird 12079	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 24895	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑠 ∈ 𝑆)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
61	60	ancom2s 650	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
62		xmetsym 24373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
63	24, 37, 32, 62	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
64	61, 63	breqtrd 5174	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡))
65	12, 2, 3, 5, 4, 7	metnrmlem1 24895	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡))
66	40	rpxrd 13076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ^*)
67	45	rpxrd 13076	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ^*)
68		xmetcl 24357	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋) → (𝑠𝐷𝑡) ∈ ℝ^*)
69	24, 32, 37, 68	syl3anc 1370	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑠𝐷𝑡) ∈ ℝ^*)
70		xle2add 13298	. . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . 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 699	. . . . . . . . . . . . . . 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 11288	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℝ)
74	73	recnd 11287	. . . . . . . . . . . . . . . . 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 12043	. . . . . . . . . . . . . . . 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 12338	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
77	73	rehalfcld 12511	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ)
78		rexmul 13310	. . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . 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 13273	. . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . 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 13338	. . . . . . . . . . . . . . . 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 5180	. . . . . . . . . . . . . 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 11309	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ^*)
86		2rp 13037	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
87	86	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℝ⁺)
88		xlemul2 13330	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 5170	. . . . . . . . . . . 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 24424	. . . . . . . . . . . 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 1384	. . . . . . . . . . 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 4054	. . . . . . . . . . 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 467	. . . . . . . . 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 3144	. . . . . . . 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 5050	. . . . . . . 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 234	. . . . . . 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 4044	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
101	100	ralrimiva 3144	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
102		iunss 5050	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
104		ss0 4408	. . . 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	eqtrid 2787	. 2 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = ∅)
107		sseq2 4022	. . . 4 ⊢ (𝑧 = 𝑉 → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ 𝑉))
108		ineq1 4221	. . . . 5 ⊢ (𝑧 = 𝑉 → (𝑧 ∩ 𝑤) = (𝑉 ∩ 𝑤))
109	108	eqeq1d 2737	. . . 4 ⊢ (𝑧 = 𝑉 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑤) = ∅))
110	107, 109	3anbi13d 1437	. . 3 ⊢ (𝑧 = 𝑉 → ((𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅)))
111		sseq2 4022	. . . 4 ⊢ (𝑤 = 𝑈 → (𝑇 ⊆ 𝑤 ↔ 𝑇 ⊆ 𝑈))
112		ineq2 4222	. . . . 5 ⊢ (𝑤 = 𝑈 → (𝑉 ∩ 𝑤) = (𝑉 ∩ 𝑈))
113	112	eqeq1d 2737	. . . 4 ⊢ (𝑤 = 𝑈 → ((𝑉 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑈) = ∅))
114	111, 113	3anbi23d 1438	. . 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 1381	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝐽 ∃𝑤 ∈ 𝐽 (𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 ifcif 4531 ∪ cuni 4912 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 infcinf 9479 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℝ^*cxr 11292 < clt 11293 ≤ cle 11294 / cdiv 11918 2c2 12319 ℝ⁺crp 13032 +_𝑒 cxad 13150 ·_e cxmu 13151 ∞Metcxmet 21367 ballcbl 21369 MetOpencmopn 21372 Clsdccld 23040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-ec 8746 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-bl 21377 df-mopn 21378 df-top 22916 df-topon 22933 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045

This theorem is referenced by: metnrm 24898

W3C validator