xrsmopn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xrsmopn		Structured version Visualization version GIF version

Theorem xrsmopn 24778

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 4881	. . . 4 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ∪ (ordTop‘ ≤ ))
2		letopuni 23172	. . . 4 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
3	1, 2	sseqtrrdi 3963	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ℝ^*)
4		eqid 2736	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
5	4	rexmet 24756	. . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
6		letop 23171	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) ∈ Top
7		reex 11129	. . . . . . . . 9 ⊢ ℝ ∈ V
8		elrestr 17391	. . . . . . . . 9 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ V ∧ 𝑥 ∈ (ordTop‘ ≤ )) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
9	6, 7, 8	mp3an12 1454	. . . . . . . 8 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
10	9	ad2antrr 727	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
11		elin 3905	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥 ∩ ℝ) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ))
12	11	biimpri 228	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
13	12	adantll 715	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
14		eqid 2736	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((ordTop‘ ≤ ) ↾_t ℝ)
15	14	xrtgioo 24772	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
16		eqid 2736	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	4, 16	tgioo 24761	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	15, 17	eqtr3i 2761	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
19	18	mopni2 24458	. . . . . . 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 1469	. . . . . 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 24775	. . . . . . . . . . 11 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)
23		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
24		ressxr 11189	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
25		sseqin2 4163	. . . . . . . . . . . . 13 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
26	24, 25	mpbi 230	. . . . . . . . . . . 12 ⊢ (ℝ^* ∩ ℝ) = ℝ
27	23, 26	eleqtrrdi 2847	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (ℝ^* ∩ ℝ))
28		rpxr 12952	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
29	28	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
30	21	xrsdsre 24776	. . . . . . . . . . . . 13 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	eqcomi 2745	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = (𝐷 ↾ (ℝ × ℝ))
32	31	blres 24396	. . . . . . . . . . 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 1469	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑦(ball‘𝐷)𝑟) ∩ ℝ))
34	21	xrsblre 24777	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ^*) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
35	28, 34	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℝ ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
36	35	adantll 715	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
37		dfss2 3907	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (𝑦(ball‘𝐷)𝑟))
40	39	sseq1d 3953	. . . . . . . 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 4177	. . . . . . . . 9 ⊢ (𝑥 ∩ ℝ) ⊆ 𝑥
42		sstr 3930	. . . . . . . . 9 ⊢ (((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) ∧ (𝑥 ∩ ℝ) ⊆ 𝑥) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
43	41, 42	mpan2 692	. . . . . . . 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 3150	. . . . . 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 12946	. . . . . 6 ⊢ 1 ∈ ℝ⁺
48	3	sselda 3921	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℝ^*)
49	48	adantr 480	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		rpxr 12952	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
51	47, 50	mp1i 13	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 1 ∈ ℝ^*)
52		elbl 24353	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 1 ∈ ℝ^) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^ ∧ (𝑦𝐷𝑧) < 1)))
53	22, 49, 51, 52	mp3an2i 1469	. . . . . . . 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 1230	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ ℝ^*)
58		xmetcl 24296	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^) → (𝑦𝐷𝑧) ∈ ℝ^)
59	22, 56, 57, 58	mp3an2i 1469	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ^*)
60		1red 11145	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 1 ∈ ℝ)
61		xmetge0 24309	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^*) → 0 ≤ (𝑦𝐷𝑧))
62	22, 56, 57, 61	mp3an2i 1469	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 0 ≤ (𝑦𝐷𝑧))
63		simpl3r 1231	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) < 1)
64		1xr 11204	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
65		xrltle 13100	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ^*) → ((𝑦𝐷𝑧) < 1 → (𝑦𝐷𝑧) ≤ 1))
66	59, 64, 65	sylancl 587	. . . . . . . . . . . . . . . . 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 13126	. . . . . . . . . . . . . . . 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 21394	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^* ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
72	56, 57, 70, 71	syl3anc 1374	. . . . . . . . . . . . . . 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 2950	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑦 = 𝑧))
77		simp1r 1200	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ 𝑥)
78		elequ1 2121	. . . . . . . . . . . 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 3927	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑦(ball‘𝐷)1) ⊆ 𝑥)
85		oveq2 7375	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)1))
86	85	sseq1d 3953	. . . . . . 7 ⊢ (𝑟 = 1 → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ (𝑦(ball‘𝐷)1) ⊆ 𝑥))
87	86	rspcev 3564	. . . . . 6 ⊢ ((1 ∈ ℝ⁺ ∧ (𝑦(ball‘𝐷)1) ⊆ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
88	47, 84, 87	sylancr 588	. . . . 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 3129	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
91		xrsmopn.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷)
92	91	elmopn2 24410	. . . 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 584	. 2 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ∈ 𝐽)
95	94	ssriv 3925	1 ⊢ (ordTop‘ ≤ ) ⊆ 𝐽

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 ∩ cin 3888 ⊆ wss 3889 ∪ cuni 4850 class class class wbr 5085 × cxp 5629 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 ℝ⁺crp 12942 (,)cioo 13298 abscabs 15196 distcds 17229 ↾_t crest 17383 topGenctg 17400 ordTopcordt 17463 ℝ^*_𝑠cxrs 17464 ∞Metcxmet 21337 ballcbl 21339 MetOpencmopn 21342 Topctop 22858

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 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-ec 8645 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fi 9324 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-seq 13964 df-exp 14024 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-tset 17239 df-ple 17240 df-ds 17242 df-rest 17385 df-topgen 17406 df-ordt 17465 df-xrs 17466 df-ps 18532 df-tsr 18533 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-bases 22911

This theorem is referenced by: xmetdcn 24804

W3C validator