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Theorem xrtgioo 23414

Description: The topology on the extended reals coincides with the standard topology on the reals, when restricted to ℝ. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypothesis**
Ref	Expression
xrtgioo.1	⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t ℝ)

**Assertion**
Ref	Expression
xrtgioo	⊢ (topGen‘ran (,)) = 𝐽

Proof of Theorem xrtgioo Dummy variables 𝑎 𝑏 𝑐 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		letop 21814	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Top
2		ioof 12836	. . . . . . . . . . 11 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6514	. . . . . . . . . . 11 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . . . . . . 10 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		iooordt 21825	. . . . . . . . . . 11 ⊢ (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )
6	5	rgen2w 3151	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )
7		ffnov 7278	. . . . . . . . . 10 ⊢ ((,):(ℝ^* × ℝ^)⟶(ordTop‘ ≤ ) ↔ ((,) Fn (ℝ^ × ℝ^) ∧ ∀𝑥 ∈ ℝ^ ∀𝑦 ∈ ℝ^* (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )))
8	4, 6, 7	mpbir2an 709	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^*)⟶(ordTop‘ ≤ )
9		frn 6520	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^*)⟶(ordTop‘ ≤ ) → ran (,) ⊆ (ordTop‘ ≤ ))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (ordTop‘ ≤ )
11		tgss 21576	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ran (,) ⊆ (ordTop‘ ≤ )) → (topGen‘ran (,)) ⊆ (topGen‘(ordTop‘ ≤ )))
12	1, 10, 11	mp2an 690	. . . . . . 7 ⊢ (topGen‘ran (,)) ⊆ (topGen‘(ordTop‘ ≤ ))
13		tgtop 21581	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ Top → (topGen‘(ordTop‘ ≤ )) = (ordTop‘ ≤ ))
14	1, 13	ax-mp 5	. . . . . . 7 ⊢ (topGen‘(ordTop‘ ≤ )) = (ordTop‘ ≤ )
15	12, 14	sseqtri 4003	. . . . . 6 ⊢ (topGen‘ran (,)) ⊆ (ordTop‘ ≤ )
16	15	sseli 3963	. . . . 5 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ∈ (ordTop‘ ≤ ))
17		retopon 23372	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
18		toponss 21535	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝑥 ∈ (topGen‘ran (,))) → 𝑥 ⊆ ℝ)
19	17, 18	mpan 688	. . . . 5 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ⊆ ℝ)
20		reordt 21826	. . . . . 6 ⊢ ℝ ∈ (ordTop‘ ≤ )
21		restopn2 21785	. . . . . 6 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ (ordTop‘ ≤ )) → (𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ↔ (𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑥 ⊆ ℝ)))
22	1, 20, 21	mp2an 690	. . . . 5 ⊢ (𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ↔ (𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑥 ⊆ ℝ))
23	16, 19, 22	sylanbrc 585	. . . 4 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
24	23	ssriv 3971	. . 3 ⊢ (topGen‘ran (,)) ⊆ ((ordTop‘ ≤ ) ↾_t ℝ)
25		eqid 2821	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
26		eqid 2821	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
27		eqid 2821	. . . . . . 7 ⊢ ran (,) = ran (,)
28	25, 26, 27	leordtval 21821	. . . . . 6 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
29	28	oveq1i 7166	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ↾_t ℝ)
30	28, 1	eqeltrri 2910	. . . . . . 7 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
31		tgclb 21578	. . . . . . 7 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases ↔ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top)
32	30, 31	mpbir 233	. . . . . 6 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases
33		reex 10628	. . . . . 6 ⊢ ℝ ∈ V
34		tgrest 21767	. . . . . 6 ⊢ ((((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases ∧ ℝ ∈ V) → (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ)) = ((topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ↾_t ℝ))
35	32, 33, 34	mp2an 690	. . . . 5 ⊢ (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ)) = ((topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ↾_t ℝ)
36	29, 35	eqtr4i 2847	. . . 4 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ))
37		retopbas 23369	. . . . 5 ⊢ ran (,) ∈ TopBases
38		elrest 16701	. . . . . . . 8 ⊢ ((((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases ∧ ℝ ∈ V) → (𝑢 ∈ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) ↔ ∃𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))𝑢 = (𝑣 ∩ ℝ)))
39	32, 33, 38	mp2an 690	. . . . . . 7 ⊢ (𝑢 ∈ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) ↔ ∃𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))𝑢 = (𝑣 ∩ ℝ))
40		elun 4125	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↔ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∨ 𝑣 ∈ ran (,)))
41		elun 4125	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ↔ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∨ 𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))))
42		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
43	42	elrnmpt 5828	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞)))
44	43	elv 3499	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞))
45		simpl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ^*)
46		pnfxr 10695	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ +∞ ∈ ℝ^*
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → +∞ ∈ ℝ^*)
48		rexr 10687	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
49	48	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		df-ioc 12744	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
51	50	elixx3g 12752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝑥(,]+∞) ↔ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
52	51	baib 538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
53	45, 47, 49, 52	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
54		pnfge 12526	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
55	49, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ +∞)
56	55	biantrud 534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
57		ltpnf 12516	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → 𝑦 < +∞)
58	57	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 < +∞)
59	58	biantrud 534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∧ 𝑦 < +∞)))
60	53, 56, 59	3bitr2d 309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 < +∞)))
61	60	pm5.32da 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ^* → ((𝑦 ∈ ℝ ∧ 𝑦 ∈ (𝑥(,]+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝑥 < 𝑦 ∧ 𝑦 < +∞))))
62		elin 4169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ (𝑥(,]+∞) ∧ 𝑦 ∈ ℝ))
63	62	biancomi 465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ (𝑥(,]+∞)))
64		3anass 1091	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞) ↔ (𝑦 ∈ ℝ ∧ (𝑥 < 𝑦 ∧ 𝑦 < +∞)))
65	61, 63, 64	3bitr4g 316	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
66		elioo2 12780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑦 ∈ (𝑥(,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
67	46, 66	mpan2 689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ (𝑥(,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
68	65, 67	bitr4d 284	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ 𝑦 ∈ (𝑥(,)+∞)))
69	68	eqrdv 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → ((𝑥(,]+∞) ∩ ℝ) = (𝑥(,)+∞))
70		ioorebas 12840	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(,)+∞) ∈ ran (,)
71	69, 70	eqeltrdi 2921	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → ((𝑥(,]+∞) ∩ ℝ) ∈ ran (,))
72		ineq1 4181	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) = ((𝑥(,]+∞) ∩ ℝ))
73	72	eleq1d 2897	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑥(,]+∞) → ((𝑣 ∩ ℝ) ∈ ran (,) ↔ ((𝑥(,]+∞) ∩ ℝ) ∈ ran (,)))
74	71, 73	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) ∈ ran (,)))
75	74	rexlimiv 3280	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) ∈ ran (,))
76	44, 75	sylbi 219	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) → (𝑣 ∩ ℝ) ∈ ran (,))
77		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
78	77	elrnmpt 5828	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥)))
79	78	elv 3499	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥))
80		mnfxr 10698	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ -∞ ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ ∈ ℝ^*)
82		df-ico 12745	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
83	82	elixx3g 12752	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (-∞[,)𝑥) ↔ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
84	83	baib 538	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
85	81, 45, 49, 84	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
86		mnfle 12530	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
87	49, 86	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ ≤ 𝑦)
88	87	biantrurd 535	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
89		mnflt 12519	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → -∞ < 𝑦)
90	89	adantl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ < 𝑦)
91	90	biantrurd 535	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 ↔ (-∞ < 𝑦 ∧ 𝑦 < 𝑥)))
92	85, 88, 91	3bitr2d 309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ < 𝑦 ∧ 𝑦 < 𝑥)))
93	92	pm5.32da 581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ^* → ((𝑦 ∈ ℝ ∧ 𝑦 ∈ (-∞[,)𝑥)) ↔ (𝑦 ∈ ℝ ∧ (-∞ < 𝑦 ∧ 𝑦 < 𝑥))))
94		elin 4169	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ (-∞[,)𝑥) ∧ 𝑦 ∈ ℝ))
95	94	biancomi 465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ (-∞[,)𝑥)))
96		3anass 1091	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ ∧ (-∞ < 𝑦 ∧ 𝑦 < 𝑥)))
97	93, 95, 96	3bitr4g 316	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
98		elioo2 12780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 ∈ (-∞(,)𝑥) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
99	80, 98	mpan 688	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ (-∞(,)𝑥) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
100	97, 99	bitr4d 284	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ 𝑦 ∈ (-∞(,)𝑥)))
101	100	eqrdv 2819	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → ((-∞[,)𝑥) ∩ ℝ) = (-∞(,)𝑥))
102		ioorebas 12840	. . . . . . . . . . . . . . . . 17 ⊢ (-∞(,)𝑥) ∈ ran (,)
103	101, 102	eqeltrdi 2921	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → ((-∞[,)𝑥) ∩ ℝ) ∈ ran (,))
104		ineq1 4181	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) = ((-∞[,)𝑥) ∩ ℝ))
105	104	eleq1d 2897	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (-∞[,)𝑥) → ((𝑣 ∩ ℝ) ∈ ran (,) ↔ ((-∞[,)𝑥) ∩ ℝ) ∈ ran (,)))
106	103, 105	syl5ibrcom 249	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) ∈ ran (,)))
107	106	rexlimiv 3280	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) ∈ ran (,))
108	79, 107	sylbi 219	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) → (𝑣 ∩ ℝ) ∈ ran (,))
109	76, 108	jaoi 853	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∨ 𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) → (𝑣 ∩ ℝ) ∈ ran (,))
110	41, 109	sylbi 219	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) → (𝑣 ∩ ℝ) ∈ ran (,))
111		elssuni 4868	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (,) → 𝑣 ⊆ ∪ ran (,))
112		unirnioo 12838	. . . . . . . . . . . . . 14 ⊢ ℝ = ∪ ran (,)
113	111, 112	sseqtrrdi 4018	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (,) → 𝑣 ⊆ ℝ)
114		df-ss 3952	. . . . . . . . . . . . 13 ⊢ (𝑣 ⊆ ℝ ↔ (𝑣 ∩ ℝ) = 𝑣)
115	113, 114	sylib 220	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran (,) → (𝑣 ∩ ℝ) = 𝑣)
116		id 22	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran (,) → 𝑣 ∈ ran (,))
117	115, 116	eqeltrd 2913	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran (,) → (𝑣 ∩ ℝ) ∈ ran (,))
118	110, 117	jaoi 853	. . . . . . . . . 10 ⊢ ((𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∨ 𝑣 ∈ ran (,)) → (𝑣 ∩ ℝ) ∈ ran (,))
119	40, 118	sylbi 219	. . . . . . . . 9 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) → (𝑣 ∩ ℝ) ∈ ran (,))
120		eleq1 2900	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∩ ℝ) → (𝑢 ∈ ran (,) ↔ (𝑣 ∩ ℝ) ∈ ran (,)))
121	119, 120	syl5ibrcom 249	. . . . . . . 8 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) → (𝑢 = (𝑣 ∩ ℝ) → 𝑢 ∈ ran (,)))
122	121	rexlimiv 3280	. . . . . . 7 ⊢ (∃𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))𝑢 = (𝑣 ∩ ℝ) → 𝑢 ∈ ran (,))
123	39, 122	sylbi 219	. . . . . 6 ⊢ (𝑢 ∈ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) → 𝑢 ∈ ran (,))
124	123	ssriv 3971	. . . . 5 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) ⊆ ran (,)
125		tgss 21576	. . . . 5 ⊢ ((ran (,) ∈ TopBases ∧ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) ⊆ ran (,)) → (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ)) ⊆ (topGen‘ran (,)))
126	37, 124, 125	mp2an 690	. . . 4 ⊢ (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ)) ⊆ (topGen‘ran (,))
127	36, 126	eqsstri 4001	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) ⊆ (topGen‘ran (,))
128	24, 127	eqssi 3983	. 2 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
129		xrtgioo.1	. 2 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t ℝ)
130	128, 129	eqtr4i 2847	1 ⊢ (topGen‘ran (,)) = 𝐽

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 class class class wbr 5066 ↦ cmpt 5146 × cxp 5553 ran crn 5556 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 +∞cpnf 10672 -∞cmnf 10673 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 (,)cioo 12739 (,]cioc 12740 [,)cico 12741 ↾_t crest 16694 topGenctg 16711 ordTopcordt 16772 Topctop 21501 TopOnctopon 21518 TopBasesctb 21553

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-rest 16696 df-topgen 16717 df-ordt 16774 df-ps 17810 df-tsr 17811 df-top 21502 df-topon 21519 df-bases 21554

This theorem is referenced by: xrrest 23415 xrsmopn 23420 xrge0tsms 23442 metdcn2 23447 xrge0tsmsd 30692 xrtgcntopre 41804 xrtgioo2 41897

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