xrsmopn - Metamath Proof Explorer

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Theorem xrsmopn 24834

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 4937	. . . 4 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ∪ (ordTop‘ ≤ ))
2		letopuni 23215	. . . 4 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
3	1, 2	sseqtrrdi 4025	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ℝ^*)
4		eqid 2737	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
5	4	rexmet 24812	. . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
6		letop 23214	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) ∈ Top
7		reex 11246	. . . . . . . . 9 ⊢ ℝ ∈ V
8		elrestr 17473	. . . . . . . . 9 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ V ∧ 𝑥 ∈ (ordTop‘ ≤ )) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
9	6, 7, 8	mp3an12 1453	. . . . . . . 8 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
10	9	ad2antrr 726	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
11		elin 3967	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥 ∩ ℝ) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ))
12	11	biimpri 228	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
13	12	adantll 714	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
14		eqid 2737	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((ordTop‘ ≤ ) ↾_t ℝ)
15	14	xrtgioo 24828	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
16		eqid 2737	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	4, 16	tgioo 24817	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	15, 17	eqtr3i 2767	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
19	18	mopni2 24506	. . . . . . 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 1468	. . . . . 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 24831	. . . . . . . . . . 11 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)
23		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
24		ressxr 11305	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
25		sseqin2 4223	. . . . . . . . . . . . 13 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
26	24, 25	mpbi 230	. . . . . . . . . . . 12 ⊢ (ℝ^* ∩ ℝ) = ℝ
27	23, 26	eleqtrrdi 2852	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (ℝ^* ∩ ℝ))
28		rpxr 13044	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
29	28	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
30	21	xrsdsre 24832	. . . . . . . . . . . . 13 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	eqcomi 2746	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = (𝐷 ↾ (ℝ × ℝ))
32	31	blres 24441	. . . . . . . . . . 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 1468	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑦(ball‘𝐷)𝑟) ∩ ℝ))
34	21	xrsblre 24833	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
35	28, 34	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
36	35	adantll 714	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
37		dfss2 3969	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ ℝ ↔ ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
38	36, 37	sylib 218	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
39	33, 38	eqtrd 2777	. . . . . . . . 9 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (𝑦(ball‘𝐷)𝑟))
40	39	sseq1d 4015	. . . . . . . 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 4237	. . . . . . . . 9 ⊢ (𝑥 ∩ ℝ) ⊆ 𝑥
42		sstr 3992	. . . . . . . . 9 ⊢ (((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) ∧ (𝑥 ∩ ℝ) ⊆ 𝑥) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
43	41, 42	mpan2 691	. . . . . . . 8 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
44	40, 43	biimtrdi 253	. . . . . . 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 3168	. . . . . 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 13038	. . . . . 6 ⊢ 1 ∈ ℝ⁺
48	3	sselda 3983	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℝ^*)
49	48	adantr 480	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		rpxr 13044	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
51	47, 50	mp1i 13	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 1 ∈ ℝ^*)
52		elbl 24398	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 1 ∈ ℝ^) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^ ∧ (𝑦𝐷𝑧) < 1)))
53	22, 49, 51, 52	mp3an2i 1468	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)))
54		simp2 1138	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → ¬ 𝑦 ∈ ℝ)
55	48	3ad2ant1 1134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ ℝ^*)
56	55	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ^*)
57		simpl3l 1229	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ ℝ^*)
58		xmetcl 24341	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^) → (𝑦𝐷𝑧) ∈ ℝ^)
59	22, 56, 57, 58	mp3an2i 1468	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ^*)
60		1red 11262	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 1 ∈ ℝ)
61		xmetge0 24354	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^*) → 0 ≤ (𝑦𝐷𝑧))
62	22, 56, 57, 61	mp3an2i 1468	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 0 ≤ (𝑦𝐷𝑧))
63		simpl3r 1230	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) < 1)
64		1xr 11320	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
65		xrltle 13191	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑦𝐷𝑧) < 1 → (𝑦𝐷𝑧) ≤ 1))
66	59, 64, 65	sylancl 586	. . . . . . . . . . . . . . . . 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 13216	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ) ∧ (0 ≤ (𝑦𝐷𝑧) ∧ (𝑦𝐷𝑧) ≤ 1)) → (𝑦𝐷𝑧) ∈ ℝ)
69	59, 60, 62, 67, 68	syl22anc 839	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ)
70		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ≠ 𝑧)
71	21	xrsdsreclb 21431	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^* ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
72	56, 57, 70, 71	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
73	69, 72	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
74	73	simpld 494	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ)
75	74	ex 412	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (𝑦 ≠ 𝑧 → 𝑦 ∈ ℝ))
76	75	necon1bd 2958	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑦 = 𝑧))
77		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ 𝑥)
78		elequ1 2115	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
79	77, 78	syl5ibcom 245	. . . . . . . . . . 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 1122	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1) → 𝑧 ∈ 𝑥))
83	53, 82	sylbid 240	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) → 𝑧 ∈ 𝑥))
84	83	ssrdv 3989	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑦(ball‘𝐷)1) ⊆ 𝑥)
85		oveq2 7439	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)1))
86	85	sseq1d 4015	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ (𝑦(ball‘𝐷)1) ⊆ 𝑥))
87	86	rspcev 3622	. . . . . 6 ⊢ ((1 ∈ ℝ⁺ ∧ (𝑦(ball‘𝐷)1) ⊆ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
88	47, 84, 87	sylancr 587	. . . . 5 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
89	46, 88	pm2.61dan 813	. . . 4 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
90	89	ralrimiva 3146	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
91		xrsmopn.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷)
92	91	elmopn2 24455	. . . 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 583	. 2 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ∈ 𝐽)
95	94	ssriv 3987	1 ⊢ (ordTop‘ ≤ ) ⊆ 𝐽

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 ℝ⁺crp 13034 (,)cioo 13387 abscabs 15273 distcds 17306 ↾_t crest 17465 topGenctg 17482 ordTopcordt 17544 ℝ^*_𝑠cxrs 17545 ∞Metcxmet 21349 ballcbl 21351 MetOpencmopn 21354 Topctop 22899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-rest 17467 df-topgen 17488 df-ordt 17546 df-xrs 17547 df-ps 18611 df-tsr 18612 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953

This theorem is referenced by: xmetdcn 24860

W3C validator