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Theorem metnrmlem3 24777

Description: Lemma for metnrm 24778. (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 4156	. . . . 5 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
7		metnrmlem.4	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
8	6, 7	eqtrid 2778	. . . 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 24776	. . 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 24776	. . 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 4163	. . . 4 ⊢ (𝑉 ∩ 𝑈) = (∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
19		iunin1 5018	. . . 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 2757	. . 3 ⊢ (𝑉 ∩ 𝑈) = ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
21	13	ineq2i 4164	. . . . . . . 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 5017	. . . . . . . 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 2757	. . . . . . 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 2731	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
26	25	cldss 22944	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
27	5, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
28	2	mopnuni 24356	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	3, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
30	27, 29	sseqtrrd 3967	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
31	30	sselda 3929	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑋)
32	31	adantrr 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑠 ∈ 𝑋)
33	25	cldss 22944	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
34	4, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
35	34, 29	sseqtrrd 3967	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
36	35	sselda 3929	. . . . . . . . . . . . 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 24774	. . . . . . . . . . . . . . . 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 12946	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ⁺)
42	41	rpxrd 12935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ^*)
43	12, 2, 3, 5, 4, 7	metnrmlem1a 24774	. . . . . . . . . . . . . . . 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 12946	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ⁺)
47	46	rpxrd 12935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ^*)
48	40	rpred 12934	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ)
49	48	rehalfcld 12368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ)
50	45	rpred 12934	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ)
51	50	rehalfcld 12368	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ)
52	49, 51	rexaddd 13133	. . . . . . . . . . . . . 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 11140	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℂ)
54	50	recnd 11140	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℂ)
55		2cnd 12203	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℂ)
56		2ne0 12229	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
57	56	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ≠ 0)
58	53, 54, 55, 57	divdird 11935	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . 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 24775	. . . . . . . . . . . . . . . . . 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 24262	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
63	24, 37, 32, 62	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
64	61, 63	breqtrd 5115	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡))
65	12, 2, 3, 5, 4, 7	metnrmlem1 24775	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡))
66	40	rpxrd 12935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ^*)
67	45	rpxrd 12935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ^*)
68		xmetcl 24246	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋) → (𝑠𝐷𝑡) ∈ ℝ^*)
69	24, 32, 37, 68	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑠𝐷𝑡) ∈ ℝ^*)
70		xle2add 13158	. . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . 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 11141	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℝ)
74	73	recnd 11140	. . . . . . . . . . . . . . . . 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 11899	. . . . . . . . . . . . . . . 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 12199	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
77	73	rehalfcld 12368	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ)
78		rexmul 13170	. . . . . . . . . . . . . . . . 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 13133	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 13198	. . . . . . . . . . . . . . . 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 5121	. . . . . . . . . . . . . 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 11162	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ^*)
86		2rp 12895	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
87	86	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℝ⁺)
88		xlemul2 13190	. . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 5111	. . . . . . . . . . . 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 24313	. . . . . . . . . . . 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 1387	. . . . . . . . . . 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 3988	. . . . . . . . . . 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 3124	. . . . . . . 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 4992	. . . . . . . 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 3968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
101	100	ralrimiva 3124	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
102		iunss 4992	. . . . 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 4349	. . . 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 2778	. 2 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = ∅)
107		sseq2 3956	. . . 4 ⊢ (𝑧 = 𝑉 → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ 𝑉))
108		ineq1 4160	. . . . 5 ⊢ (𝑧 = 𝑉 → (𝑧 ∩ 𝑤) = (𝑉 ∩ 𝑤))
109	108	eqeq1d 2733	. . . 4 ⊢ (𝑧 = 𝑉 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑤) = ∅))
110	107, 109	3anbi13d 1440	. . 3 ⊢ (𝑧 = 𝑉 → ((𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅)))
111		sseq2 3956	. . . 4 ⊢ (𝑤 = 𝑈 → (𝑇 ⊆ 𝑤 ↔ 𝑇 ⊆ 𝑈))
112		ineq2 4161	. . . . 5 ⊢ (𝑤 = 𝑈 → (𝑉 ∩ 𝑤) = (𝑉 ∩ 𝑈))
113	112	eqeq1d 2733	. . . 4 ⊢ (𝑤 = 𝑈 → ((𝑉 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑈) = ∅))
114	111, 113	3anbi23d 1441	. . 3 ⊢ (𝑤 = 𝑈 → ((𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈 ∧ (𝑉 ∩ 𝑈) = ∅)))
115	110, 114	rspc2ev 3585	. 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 1384	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 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ∩ cin 3896 ⊆ wss 3897 ∅c0 4280 ifcif 4472 ∪ cuni 4856 ∪ ciun 4939 class class class wbr 5089 ↦ cmpt 5170 ran crn 5615 ‘cfv 6481 (class class class)co 7346 infcinf 9325 ℝcr 11005 0cc0 11006 1c1 11007 + caddc 11009 · cmul 11011 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 / cdiv 11774 2c2 12180 ℝ⁺crp 12890 +_𝑒 cxad 13009 ·_e cxmu 13010 ∞Metcxmet 21276 ballcbl 21278 MetOpencmopn 21281 Clsdccld 22931

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-ec 8624 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-icc 13252 df-topgen 17347 df-psmet 21283 df-xmet 21284 df-bl 21286 df-mopn 21287 df-top 22809 df-topon 22826 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936

This theorem is referenced by: metnrm 24778

W3C validator