xrsmopn - Metamath Proof Explorer

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Theorem xrsmopn 24175

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 4898	. . . 4 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ∪ (ordTop‘ ≤ ))
2		letopuni 22558	. . . 4 ⊢ ℝ^* = ∪ (ordTop‘ ≤ )
3	1, 2	sseqtrrdi 3995	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → 𝑥 ⊆ ℝ^*)
4		eqid 2736	. . . . . . . 8 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
5	4	rexmet 24154	. . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (∞Met‘ℝ)
6		letop 22557	. . . . . . . . 9 ⊢ (ordTop‘ ≤ ) ∈ Top
7		reex 11142	. . . . . . . . 9 ⊢ ℝ ∈ V
8		elrestr 17310	. . . . . . . . 9 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ V ∧ 𝑥 ∈ (ordTop‘ ≤ )) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
9	6, 7, 8	mp3an12 1451	. . . . . . . 8 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
10	9	ad2antrr 724	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → (𝑥 ∩ ℝ) ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
11		elin 3926	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑥 ∩ ℝ) ↔ (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ))
12	11	biimpri 227	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
13	12	adantll 712	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ (𝑥 ∩ ℝ))
14		eqid 2736	. . . . . . . . . 10 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((ordTop‘ ≤ ) ↾_t ℝ)
15	14	xrtgioo 24169	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
16		eqid 2736	. . . . . . . . . 10 ⊢ (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ))) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
17	4, 16	tgioo 24159	. . . . . . . . 9 ⊢ (topGen‘ran (,)) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
18	15, 17	eqtr3i 2766	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (MetOpen‘((abs ∘ − ) ↾ (ℝ × ℝ)))
19	18	mopni2 23849	. . . . . . 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 1466	. . . . . 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 24172	. . . . . . . . . . 11 ⊢ 𝐷 ∈ (∞Met‘ℝ^*)
23		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ ℝ)
24		ressxr 11199	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℝ^*
25		sseqin2 4175	. . . . . . . . . . . . 13 ⊢ (ℝ ⊆ ℝ^* ↔ (ℝ^* ∩ ℝ) = ℝ)
26	24, 25	mpbi 229	. . . . . . . . . . . 12 ⊢ (ℝ^* ∩ ℝ) = ℝ
27	23, 26	eleqtrrdi 2849	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ (ℝ^* ∩ ℝ))
28		rpxr 12924	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
29	28	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
30	21	xrsdsre 24173	. . . . . . . . . . . . 13 ⊢ (𝐷 ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ))
31	30	eqcomi 2745	. . . . . . . . . . . 12 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = (𝐷 ↾ (ℝ × ℝ))
32	31	blres 23784	. . . . . . . . . . 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 1466	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = ((𝑦(ball‘𝐷)𝑟) ∩ ℝ))
34	21	xrsblre 24174	. . . . . . . . . . . . 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 712	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘𝐷)𝑟) ⊆ ℝ)
37		df-ss 3927	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ ℝ ↔ ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
38	36, 37	sylib 217	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → ((𝑦(ball‘𝐷)𝑟) ∩ ℝ) = (𝑦(ball‘𝐷)𝑟))
39	33, 38	eqtrd 2776	. . . . . . . . 9 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ 𝑦 ∈ ℝ) ∧ 𝑟 ∈ ℝ⁺) → (𝑦(ball‘((abs ∘ − ) ↾ (ℝ × ℝ)))𝑟) = (𝑦(ball‘𝐷)𝑟))
40	39	sseq1d 3975	. . . . . . . 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 3952	. . . . . . . . 9 ⊢ (((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) ∧ (𝑥 ∩ ℝ) ⊆ 𝑥) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
43	41, 42	mpan2 689	. . . . . . . 8 ⊢ ((𝑦(ball‘𝐷)𝑟) ⊆ (𝑥 ∩ ℝ) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
44	40, 43	syl6bi 252	. . . . . . 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 3165	. . . . . 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 12919	. . . . . 6 ⊢ 1 ∈ ℝ⁺
48	3	sselda 3944	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ ℝ^*)
49	48	adantr 481	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		rpxr 12924	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → 1 ∈ ℝ^*)
51	47, 50	mp1i 13	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → 1 ∈ ℝ^*)
52		elbl 23741	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 1 ∈ ℝ^) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^ ∧ (𝑦𝐷𝑧) < 1)))
53	22, 49, 51, 52	mp3an2i 1466	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) ↔ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)))
54		simp2 1137	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → ¬ 𝑦 ∈ ℝ)
55	48	3ad2ant1 1133	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ ℝ^*)
56	55	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ^*)
57		simpl3l 1228	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑧 ∈ ℝ^*)
58		xmetcl 23684	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^) → (𝑦𝐷𝑧) ∈ ℝ^)
59	22, 56, 57, 58	mp3an2i 1466	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ^*)
60		1red 11156	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 1 ∈ ℝ)
61		xmetge0 23697	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐷 ∈ (∞Met‘ℝ^) ∧ 𝑦 ∈ ℝ^ ∧ 𝑧 ∈ ℝ^*) → 0 ≤ (𝑦𝐷𝑧))
62	22, 56, 57, 61	mp3an2i 1466	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 0 ≤ (𝑦𝐷𝑧))
63		simpl3r 1229	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) < 1)
64		1xr 11214	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℝ^*
65		xrltle 13068	. . . . . . . . . . . . . . . . . 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 13093	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦𝐷𝑧) ∈ ℝ^* ∧ 1 ∈ ℝ) ∧ (0 ≤ (𝑦𝐷𝑧) ∧ (𝑦𝐷𝑧) ≤ 1)) → (𝑦𝐷𝑧) ∈ ℝ)
69	59, 60, 62, 67, 68	syl22anc 837	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦𝐷𝑧) ∈ ℝ)
70		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ≠ 𝑧)
71	21	xrsdsreclb 20844	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℝ^* ∧ 𝑧 ∈ ℝ^* ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
72	56, 57, 70, 71	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → ((𝑦𝐷𝑧) ∈ ℝ ↔ (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ)))
73	69, 72	mpbid 231	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → (𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ))
74	73	simpld 495	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) ∧ 𝑦 ≠ 𝑧) → 𝑦 ∈ ℝ)
75	74	ex 413	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (𝑦 ≠ 𝑧 → 𝑦 ∈ ℝ))
76	75	necon1bd 2961	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → (¬ 𝑦 ∈ ℝ → 𝑦 = 𝑧))
77		simp1r 1198	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ ∧ (𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1)) → 𝑦 ∈ 𝑥)
78		elequ1 2113	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥))
79	77, 78	syl5ibcom 244	. . . . . . . . . . 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 1121	. . . . . . . 8 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ((𝑧 ∈ ℝ^* ∧ (𝑦𝐷𝑧) < 1) → 𝑧 ∈ 𝑥))
83	53, 82	sylbid 239	. . . . . . 7 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑧 ∈ (𝑦(ball‘𝐷)1) → 𝑧 ∈ 𝑥))
84	83	ssrdv 3950	. . . . . 6 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → (𝑦(ball‘𝐷)1) ⊆ 𝑥)
85		oveq2 7365	. . . . . . . 8 ⊢ (𝑟 = 1 → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘𝐷)1))
86	85	sseq1d 3975	. . . . . . 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 587	. . . . 5 ⊢ (((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) ∧ ¬ 𝑦 ∈ ℝ) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
89	46, 88	pm2.61dan 811	. . . 4 ⊢ ((𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑦 ∈ 𝑥) → ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
90	89	ralrimiva 3143	. . 3 ⊢ (𝑥 ∈ (ordTop‘ ≤ ) → ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ⁺ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)
91		xrsmopn.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷)
92	91	elmopn2 23798	. . . 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 3948	1 ⊢ (ordTop‘ ≤ ) ⊆ 𝐽

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∪ cuni 4865 class class class wbr 5105 × cxp 5631 ran crn 5634 ↾ cres 5635 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 ℝcr 11050 0cc0 11051 1c1 11052 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 ℝ⁺crp 12915 (,)cioo 13264 abscabs 15119 distcds 17142 ↾_t crest 17302 topGenctg 17319 ordTopcordt 17381 ℝ^*_𝑠cxrs 17382 ∞Metcxmet 20781 ballcbl 20783 MetOpencmopn 20786 Topctop 22242

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-ec 8650 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-tset 17152 df-ple 17153 df-ds 17155 df-rest 17304 df-topgen 17325 df-ordt 17383 df-xrs 17384 df-ps 18455 df-tsr 18456 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-bases 22296

This theorem is referenced by: xmetdcn 24201

W3C validator