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Theorem xrsmopn 25042

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 4899	. . . 4 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ∪ (ordTop‘ ≤ ))
2		letopuni 23435	. . . 4 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
3	1, 2	sseqtrrdi 3972	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ℝ^*)
4		eqid 2760	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
5	4	rexmet 25020	. . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
6		letop 23434	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) ∈ Top
7		reex 11218	. . . . . . . . 9 ⊢ ℝ ∈ V
8		elrestr 17516	. . . . . . . . 9 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ V ∧ 𝑥 ∈ (ordTop‘ ≤ )) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
9	6, 7, 8	mp3an12 1480	. . . . . . . 8 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
10	9	ad2antrr 739	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
11		elin 3915	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥 ∩ ℝ) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ))
12	11	biimpri 231	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
13	12	adantll 727	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
14		eqid 2760	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((ordTop‘ ≤ ) ↾_t ℝ)
15	14	xrtgioo 25036	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
16		eqid 2760	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	4, 16	tgioo 25025	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	15, 17	eqtr3i 2785	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
19	18	mopni2 24722	. . . . . . 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 1495	. . . . . 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 25039	. . . . . . . . . . 11 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)
23		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
24		ressxr 11280	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
25		sseqin2 4169	. . . . . . . . . . . . 13 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
26	24, 25	mpbi 233	. . . . . . . . . . . 12 ⊢ (ℝ^* ∩ ℝ) = ℝ
27	23, 26	eleqtrrdi 2871	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (ℝ^* ∩ ℝ))
28		rpxr 13055	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
29	28	adantl 487	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
30	21	xrsdsre 25040	. . . . . . . . . . . . 13 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	eqcomi 2769	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = (𝐷 ↾ (ℝ × ℝ))
32	31	blres 24660	. . . . . . . . . . 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 1495	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑦(ball‘𝐷)𝑟) ∩ ℝ))
34	21	xrsblre 25041	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
35	28, 34	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
36	35	adantll 727	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
37		dfss2 3917	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ ℝ ↔ ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
38	36, 37	sylib 221	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
39	33, 38	eqtrd 2795	. . . . . . . . 9 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (𝑦(ball‘𝐷)𝑟))
40	39	sseq1d 3962	. . . . . . . 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 4182	. . . . . . . . 9 ⊢ (𝑥 ∩ ℝ) ⊆ 𝑥
42		sstr 3939	. . . . . . . . 9 ⊢ (((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) ∧ (𝑥 ∩ ℝ) ⊆ 𝑥) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
43	41, 42	mpan2 704	. . . . . . . 8 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
44	40, 43	biimtrdi 256	. . . . . . 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 16	. . . . 5 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
47		1rp 13049	. . . . . 6 ⊢ 1 ∈ ℝ⁺
48	3	sselda 3931	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℝ^*)
49	48	adantr 486	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		rpxr 13055	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
51	47, 50	mp1i 14	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 1 ∈ ℝ^*)
52		elbl 24617	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 1 ∈ ℝ^) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^ ∧ (𝑦𝐷𝑧) < 1)))
53	22, 49, 51, 52	mp3an2i 1495	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)))
54		simp2 1155	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → ¬ 𝑦 ∈ ℝ)
55	48	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ ℝ^*)
56	55	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ^*)
57		simpl3l 1247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ ℝ^*)
58		xmetcl 24560	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^) → (𝑦𝐷𝑧) ∈ ℝ^)
59	22, 56, 57, 58	mp3an2i 1495	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ^*)
60		1red 11236	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 1 ∈ ℝ)
61		xmetge0 24573	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^*) → 0 ≤ (𝑦𝐷𝑧))
62	22, 56, 57, 61	mp3an2i 1495	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 0 ≤ (𝑦𝐷𝑧))
63		simpl3r 1248	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) < 1)
64		1xr 11295	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
65		xrltle 13203	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑦𝐷𝑧) < 1 → (𝑦𝐷𝑧) ≤ 1))
66	59, 64, 65	sylancl 598	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) < 1 → (𝑦𝐷𝑧) ≤ 1))
67	63, 66	mpd 16	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ≤ 1)
68		xrrege0 13229	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ) ∧ (0 ≤ (𝑦𝐷𝑧) ∧ (𝑦𝐷𝑧) ≤ 1)) → (𝑦𝐷𝑧) ∈ ℝ)
69	59, 60, 62, 67, 68	syl22anc 852	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ)
70		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ≠ 𝑧)
71	21	xrsdsreclb 21630	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^* ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
72	56, 57, 70, 71	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
73	69, 72	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
74	73	simpld 500	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ)
75	74	ex 418	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (𝑦 ≠ 𝑧 → 𝑦 ∈ ℝ))
76	75	necon1bd 2973	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑦 = 𝑧))
77		simp1r 1217	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ 𝑥)
78		elequ1 2152	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
79	77, 78	syl5ibcom 248	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (𝑦 = 𝑧 → 𝑧 ∈ 𝑥))
80	76, 79	syld 48	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑧 ∈ 𝑥))
81	54, 80	mpd 16	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑧 ∈ 𝑥)
82	81	3expia 1139	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1) → 𝑧 ∈ 𝑥))
83	53, 82	sylbid 243	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) → 𝑧 ∈ 𝑥))
84	83	ssrdv 3937	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑦(ball‘𝐷)1) ⊆ 𝑥)
85		oveq2 7422	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)1))
86	85	sseq1d 3962	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ (𝑦(ball‘𝐷)1) ⊆ 𝑥))
87	86	rspcev 3576	. . . . . 6 ⊢ ((1 ∈ ℝ⁺ ∧ (𝑦(ball‘𝐷)1) ⊆ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
88	47, 84, 87	sylancr 599	. . . . 5 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
89	46, 88	pm2.61dan 825	. . . 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 24674	. . . 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 595	. 2 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ∈ 𝐽)
95	94	ssriv 3935	1 ⊢ (ordTop‘ ≤ ) ⊆ 𝐽

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 ∪ cuni 4867 class class class wbr 5103 × cxp 5653 ran crn 5656 ↾ cres 5657 ∘ ccom 5659 ‘cfv 6533 (class class class)co 7414 ℝcr 11126 0cc0 11127 1c1 11128 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 − cmin 11468 ℝ⁺crp 13045 (,)cioo 13401 abscabs 15324 distcds 17354 ↾_t crest 17508 topGenctg 17525 ordTopcordt 17588 ℝ^*_𝑠cxrs 17589 ∞Metcxmet 21573 ballcbl 21575 MetOpencmopn 21578 Topctop 23121

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8458 df-2o 8459 df-er 8699 df-ec 8701 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9384 df-sup 9415 df-inf 9416 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-div 11899 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-q 13001 df-rp 13046 df-xneg 13166 df-xadd 13167 df-xmul 13168 df-ioo 13405 df-ioc 13406 df-ico 13407 df-icc 13408 df-fz 13565 df-seq 14069 df-exp 14129 df-cj 15189 df-re 15190 df-im 15191 df-sqrt 15325 df-abs 15326 df-struct 17242 df-slot 17277 df-ndx 17289 df-base 17305 df-plusg 17358 df-mulr 17359 df-tset 17364 df-ple 17365 df-ds 17367 df-rest 17510 df-topgen 17531 df-ordt 17590 df-xrs 17591 df-ps 18657 df-tsr 18658 df-psmet 21580 df-xmet 21581 df-met 21582 df-bl 21583 df-mopn 21584 df-top 23122 df-topon 23139 df-bases 23174

This theorem is used by: xmetdcn 25068

W3C validator