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Theorem metnrmlem3 24818

Description: Lemma for metnrm 24819. (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 4163	. . . . 5 ⊢ (𝑇 ∩ 𝑆) = (𝑆 ∩ 𝑇)
7		metnrmlem.4	. . . . 5 ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
8	6, 7	eqtrid 2784	. . . 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 24817	. . 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 24817	. . 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 4170	. . . 4 ⊢ (𝑉 ∩ 𝑈) = (∪ 𝑠 ∈ 𝑆 (𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
19		iunin1 5029	. . . 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 2763	. . 3 ⊢ (𝑉 ∩ 𝑈) = ∪ 𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈)
21	13	ineq2i 4171	. . . . . . . 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 5028	. . . . . . . 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 2763	. . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝐽 = ∪ 𝐽
26	25	cldss 22985	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
27	5, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
28	2	mopnuni 24397	. . . . . . . . . . . . . . . 16 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
29	3, 28	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
30	27, 29	sseqtrrd 3973	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
31	30	sselda 3935	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑋)
32	31	adantrr 718	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑠 ∈ 𝑋)
33	25	cldss 22985	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
34	4, 33	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐽)
35	34, 29	sseqtrrd 3973	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
36	35	sselda 3935	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑋)
37	36	adantrl 717	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 𝑡 ∈ 𝑋)
38	1, 2, 3, 4, 5, 8	metnrmlem1a 24815	. . . . . . . . . . . . . . . 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 718	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ⁺)
41	40	rphalfcld 12973	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ⁺)
42	41	rpxrd 12962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ^*)
43	12, 2, 3, 5, 4, 7	metnrmlem1a 24815	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 < (𝐹‘𝑡) ∧ if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ⁺))
44	43	adantrl 717	. . . . . . . . . . . . . . 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 12973	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ⁺)
47	46	rpxrd 12962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ^*)
48	40	rpred 12961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ)
49	48	rehalfcld 12400	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2) ∈ ℝ)
50	45	rpred 12961	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ)
51	50	rehalfcld 12400	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) / 2) ∈ ℝ)
52	49, 51	rexaddd 13161	. . . . . . . . . . . . . 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 11172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℂ)
54	50	recnd 11172	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℂ)
55		2cnd 12235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℂ)
56		2ne0 12261	. . . . . . . . . . . . . . . 16 ⊢ 2 ≠ 0
57	56	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ≠ 0)
58	53, 54, 55, 57	divdird 11967	. . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . 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 24816	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝑇 ∧ 𝑠 ∈ 𝑆)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
61	60	ancom2s 651	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑡𝐷𝑠))
62		xmetsym 24303	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑡 ∈ 𝑋 ∧ 𝑠 ∈ 𝑋) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
63	24, 37, 32, 62	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑡𝐷𝑠) = (𝑠𝐷𝑡))
64	61, 63	breqtrd 5126	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ≤ (𝑠𝐷𝑡))
65	12, 2, 3, 5, 4, 7	metnrmlem1 24816	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ≤ (𝑠𝐷𝑡))
66	40	rpxrd 12962	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) ∈ ℝ^*)
67	45	rpxrd 12962	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡)) ∈ ℝ^*)
68		xmetcl 24287	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑠 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋) → (𝑠𝐷𝑡) ∈ ℝ^*)
69	24, 32, 37, 68	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (𝑠𝐷𝑡) ∈ ℝ^*)
70		xle2add 13186	. . . . . . . . . . . . . . . . 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 700	. . . . . . . . . . . . . . 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 11173	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → (if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) ∈ ℝ)
74	73	recnd 11172	. . . . . . . . . . . . . . . . 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 11931	. . . . . . . . . . . . . . . 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 12231	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∈ ℝ
77	73	rehalfcld 12400	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ)
78		rexmul 13198	. . . . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . . . 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 13161	. . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . 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 13226	. . . . . . . . . . . . . . . 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 5132	. . . . . . . . . . . . . 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 11194	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → ((if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) + if(1 ≤ (𝐹‘𝑡), 1, (𝐹‘𝑡))) / 2) ∈ ℝ^*)
86		2rp 12922	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ⁺
87	86	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝑆 ∧ 𝑡 ∈ 𝑇)) → 2 ∈ ℝ⁺)
88		xlemul2 13218	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 5122	. . . . . . . . . . . 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 24354	. . . . . . . . . . . 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 1388	. . . . . . . . . . 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 3994	. . . . . . . . . . 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 3130	. . . . . . . 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 5002	. . . . . . . 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 3974	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
101	100	ralrimiva 3130	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑠(ball‘𝐷)(if(1 ≤ (𝐺‘𝑠), 1, (𝐺‘𝑠)) / 2)) ∩ 𝑈) ⊆ ∅)
102		iunss 5002	. . . . 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 4356	. . . 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 2784	. 2 ⊢ (𝜑 → (𝑉 ∩ 𝑈) = ∅)
107		sseq2 3962	. . . 4 ⊢ (𝑧 = 𝑉 → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ 𝑉))
108		ineq1 4167	. . . . 5 ⊢ (𝑧 = 𝑉 → (𝑧 ∩ 𝑤) = (𝑉 ∩ 𝑤))
109	108	eqeq1d 2739	. . . 4 ⊢ (𝑧 = 𝑉 → ((𝑧 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑤) = ∅))
110	107, 109	3anbi13d 1441	. . 3 ⊢ (𝑧 = 𝑉 → ((𝑆 ⊆ 𝑧 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑧 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅)))
111		sseq2 3962	. . . 4 ⊢ (𝑤 = 𝑈 → (𝑇 ⊆ 𝑤 ↔ 𝑇 ⊆ 𝑈))
112		ineq2 4168	. . . . 5 ⊢ (𝑤 = 𝑈 → (𝑉 ∩ 𝑤) = (𝑉 ∩ 𝑈))
113	112	eqeq1d 2739	. . . 4 ⊢ (𝑤 = 𝑈 → ((𝑉 ∩ 𝑤) = ∅ ↔ (𝑉 ∩ 𝑈) = ∅))
114	111, 113	3anbi23d 1442	. . 3 ⊢ (𝑤 = 𝑈 → ((𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑤 ∧ (𝑉 ∩ 𝑤) = ∅) ↔ (𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈 ∧ (𝑉 ∩ 𝑈) = ∅)))
115	110, 114	rspc2ev 3591	. 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 1385	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 ifcif 4481 ∪ cuni 4865 ∪ ciun 4948 class class class wbr 5100 ↦ cmpt 5181 ran crn 5633 ‘cfv 6500 (class class class)co 7368 infcinf 9356 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 · cmul 11043 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 / cdiv 11806 2c2 12212 ℝ⁺crp 12917 +_𝑒 cxad 13036 ·_e cxmu 13037 ∞Metcxmet 21306 ballcbl 21308 MetOpencmopn 21311 Clsdccld 22972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-ec 8647 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-bl 21316 df-mopn 21317 df-top 22850 df-topon 22867 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977

This theorem is referenced by: metnrm 24819

W3C validator