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Theorem xrsmopn 24873

Description: The metric on the extended reals generates a topology, but this does not match the order topology on ℝ^*; for example {+∞} is open in the metric topology, but not the order topology. However, the metric topology is finer than the order topology, meaning that all open intervals are open in the metric topology. (Contributed by Mario Carneiro, 4-Sep-2015.)

**Hypotheses**
Ref	Expression
xrsxmet.1	⊢ 𝐷 = (dist‘ℝ^*_𝑠)
xrsmopn.1	⊢ 𝐽 = (MetOpen‘𝐷)

**Assertion**
Ref	Expression
xrsmopn	⊢ (ordTop‘ ≤ ) ⊆ 𝐽

Proof of Theorem xrsmopn Dummy variables 𝑥 𝑟 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elssuni 4897	. . . 4 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ∪ (ordTop‘ ≤ ))
2		letopuni 23267	. . . 4 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
3	1, 2	sseqtrrdi 3977	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ℝ^*)
4		eqid 2762	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
5	4	rexmet 24851	. . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
6		letop 23266	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) ∈ Top
7		reex 11164	. . . . . . . . 9 ⊢ ℝ ∈ V
8		elrestr 17457	. . . . . . . . 9 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ V ∧ 𝑥 ∈ (ordTop‘ ≤ )) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
9	6, 7, 8	mp3an12 1472	. . . . . . . 8 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
10	9	ad2antrr 736	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
11		elin 3920	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥 ∩ ℝ) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ))
12	11	biimpri 230	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
13	12	adantll 724	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
14		eqid 2762	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((ordTop‘ ≤ ) ↾_t ℝ)
15	14	xrtgioo 24867	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
16		eqid 2762	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	4, 16	tgioo 24856	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	15, 17	eqtr3i 2787	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
19	18	mopni2 24553	. . . . . . 7 ⊢ ((((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ) ∧ (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ∧ 𝑦 ∈ (𝑥 ∩ ℝ)) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ (𝑥 ∩ ℝ))
20	5, 10, 13, 19	mp3an2i 1487	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ (𝑥 ∩ ℝ))
21		xrsxmet.1	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘ℝ^*_𝑠)
22	21	xrsxmet 24870	. . . . . . . . . . 11 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)
23		simplr 778	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
24		ressxr 11226	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
25		sseqin2 4175	. . . . . . . . . . . . 13 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
26	24, 25	mpbi 232	. . . . . . . . . . . 12 ⊢ (ℝ^* ∩ ℝ) = ℝ
27	23, 26	eleqtrrdi 2873	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (ℝ^* ∩ ℝ))
28		rpxr 13003	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
29	28	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
30	21	xrsdsre 24871	. . . . . . . . . . . . 13 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	eqcomi 2771	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = (𝐷 ↾ (ℝ × ℝ))
32	31	blres 24491	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ (ℝ^ ∩ ℝ) ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑦(ball‘𝐷)𝑟) ∩ ℝ))
33	22, 27, 29, 32	mp3an2i 1487	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑦(ball‘𝐷)𝑟) ∩ ℝ))
34	21	xrsblre 24872	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
35	28, 34	sylan2 602	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
36	35	adantll 724	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
37		dfss2 3922	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ ℝ ↔ ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
38	36, 37	sylib 220	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
39	33, 38	eqtrd 2797	. . . . . . . . 9 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (𝑦(ball‘𝐷)𝑟))
40	39	sseq1d 3967	. . . . . . . 8 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ((𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ (𝑥 ∩ ℝ) ↔ (𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ)))
41		inss1 4188	. . . . . . . . 9 ⊢ (𝑥 ∩ ℝ) ⊆ 𝑥
42		sstr 3944	. . . . . . . . 9 ⊢ (((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) ∧ (𝑥 ∩ ℝ) ⊆ 𝑥) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
43	41, 42	mpan2 701	. . . . . . . 8 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
44	40, 43	biimtrdi 255	. . . . . . 7 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ((𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ (𝑥 ∩ ℝ) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥))
45	44	reximdva 3175	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → (∃𝑟 ∈ ℝ⁺ (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) ⊆ (𝑥 ∩ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥))
46	20, 45	mpd 15	. . . . 5 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
47		1rp 12997	. . . . . 6 ⊢ 1 ∈ ℝ⁺
48	3	sselda 3936	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℝ^*)
49	48	adantr 484	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		rpxr 13003	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
51	47, 50	mp1i 13	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 1 ∈ ℝ^*)
52		elbl 24448	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 1 ∈ ℝ^) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^ ∧ (𝑦𝐷𝑧) < 1)))
53	22, 49, 51, 52	mp3an2i 1487	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)))
54		simp2 1150	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → ¬ 𝑦 ∈ ℝ)
55	48	3ad2ant1 1146	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ ℝ^*)
56	55	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ^*)
57		simpl3l 1242	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ ℝ^*)
58		xmetcl 24391	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^) → (𝑦𝐷𝑧) ∈ ℝ^)
59	22, 56, 57, 58	mp3an2i 1487	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ^*)
60		1red 11182	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 1 ∈ ℝ)
61		xmetge0 24404	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^*) → 0 ≤ (𝑦𝐷𝑧))
62	22, 56, 57, 61	mp3an2i 1487	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 0 ≤ (𝑦𝐷𝑧))
63		simpl3r 1243	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) < 1)
64		1xr 11241	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
65		xrltle 13151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑦𝐷𝑧) < 1 → (𝑦𝐷𝑧) ≤ 1))
66	59, 64, 65	sylancl 595	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) < 1 → (𝑦𝐷𝑧) ≤ 1))
67	63, 66	mpd 15	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ≤ 1)
68		xrrege0 13177	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ) ∧ (0 ≤ (𝑦𝐷𝑧) ∧ (𝑦𝐷𝑧) ≤ 1)) → (𝑦𝐷𝑧) ∈ ℝ)
69	59, 60, 62, 67, 68	syl22anc 849	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ)
70		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ≠ 𝑧)
71	21	xrsdsreclb 21466	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^* ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
72	56, 57, 70, 71	syl3anc 1390	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
73	69, 72	mpbid 234	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
74	73	simpld 498	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ)
75	74	ex 416	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (𝑦 ≠ 𝑧 → 𝑦 ∈ ℝ))
76	75	necon1bd 2975	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑦 = 𝑧))
77		simp1r 1212	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ 𝑥)
78		elequ1 2149	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
79	77, 78	syl5ibcom 247	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (𝑦 = 𝑧 → 𝑧 ∈ 𝑥))
80	76, 79	syld 47	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑧 ∈ 𝑥))
81	54, 80	mpd 15	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑧 ∈ 𝑥)
82	81	3expia 1134	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1) → 𝑧 ∈ 𝑥))
83	53, 82	sylbid 242	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) → 𝑧 ∈ 𝑥))
84	83	ssrdv 3942	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑦(ball‘𝐷)1) ⊆ 𝑥)
85		oveq2 7404	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)1))
86	85	sseq1d 3967	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ (𝑦(ball‘𝐷)1) ⊆ 𝑥))
87	86	rspcev 3581	. . . . . 6 ⊢ ((1 ∈ ℝ⁺ ∧ (𝑦(ball‘𝐷)1) ⊆ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
88	47, 84, 87	sylancr 596	. . . . 5 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
89	46, 88	pm2.61dan 822	. . . 4 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
90	89	ralrimiva 3154	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
91		xrsmopn.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷)
92	91	elmopn2 24505	. . . 4 ⊢ (𝐷 ∈ (∞Met‘ℝ^) → (𝑥 ∈ 𝐽 ↔ (𝑥 ⊆ ℝ^ ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
93	22, 92	ax-mp 5	. . 3 ⊢ (𝑥 ∈ 𝐽 ↔ (𝑥 ⊆ ℝ^* ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥))
94	3, 90, 93	sylanbrc 592	. 2 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ∈ 𝐽)
95	94	ssriv 3940	1 ⊢ (ordTop‘ ≤ ) ⊆ 𝐽

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ≠ wne 2957 ∀wral 3076 ∃wrex 3086 Vcvv 3454 ∩ cin 3903 ⊆ wss 3904 ∪ cuni 4865 class class class wbr 5100 × cxp 5645 ran crn 5648 ↾ cres 5649 ∘ ccom 5651 ‘cfv 6521 (class class class)co 7396 ℝcr 11072 0cc0 11073 1c1 11074 ℝ^*cxr 11215 < clt 11216 ≤ cle 11217 − cmin 11414 ℝ⁺crp 12993 (,)cioo 13349 abscabs 15261 distcds 17295 ↾_t crest 17449 topGenctg 17466 ordTopcordt 17529 ℝ^*_𝑠cxrs 17530 ∞Metcxmet 21409 ballcbl 21411 MetOpencmopn 21414 Topctop 22953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-ec 8680 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fi 9357 df-sup 9388 df-inf 9389 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-seq 14015 df-exp 14075 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-slot 17218 df-ndx 17230 df-base 17246 df-plusg 17299 df-mulr 17300 df-tset 17305 df-ple 17306 df-ds 17308 df-rest 17451 df-topgen 17472 df-ordt 17531 df-xrs 17532 df-ps 18598 df-tsr 18599 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-top 22954 df-topon 22971 df-bases 23006

This theorem is referenced by: xmetdcn 24899

W3C validator