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Theorem xrtgioo 24314

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 22702	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Top
2		ioof 13421	. . . . . . . . . . 11 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6715	. . . . . . . . . . 11 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . . . . . . 10 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		iooordt 22713	. . . . . . . . . . 11 ⊢ (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )
6	5	rgen2w 3067	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )
7		ffnov 7532	. . . . . . . . . 10 ⊢ ((,):(ℝ^* × ℝ^)⟶(ordTop‘ ≤ ) ↔ ((,) Fn (ℝ^ × ℝ^) ∧ ∀𝑥 ∈ ℝ^ ∀𝑦 ∈ ℝ^* (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )))
8	4, 6, 7	mpbir2an 710	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^*)⟶(ordTop‘ ≤ )
9		frn 6722	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^*)⟶(ordTop‘ ≤ ) → ran (,) ⊆ (ordTop‘ ≤ ))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (ordTop‘ ≤ )
11		tgss 22463	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ran (,) ⊆ (ordTop‘ ≤ )) → (topGen‘ran (,)) ⊆ (topGen‘(ordTop‘ ≤ )))
12	1, 10, 11	mp2an 691	. . . . . . 7 ⊢ (topGen‘ran (,)) ⊆ (topGen‘(ordTop‘ ≤ ))
13		tgtop 22468	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ Top → (topGen‘(ordTop‘ ≤ )) = (ordTop‘ ≤ ))
14	1, 13	ax-mp 5	. . . . . . 7 ⊢ (topGen‘(ordTop‘ ≤ )) = (ordTop‘ ≤ )
15	12, 14	sseqtri 4018	. . . . . 6 ⊢ (topGen‘ran (,)) ⊆ (ordTop‘ ≤ )
16	15	sseli 3978	. . . . 5 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ∈ (ordTop‘ ≤ ))
17		retopon 24272	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
18		toponss 22421	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝑥 ∈ (topGen‘ran (,))) → 𝑥 ⊆ ℝ)
19	17, 18	mpan 689	. . . . 5 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ⊆ ℝ)
20		reordt 22714	. . . . . 6 ⊢ ℝ ∈ (ordTop‘ ≤ )
21		restopn2 22673	. . . . . 6 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ (ordTop‘ ≤ )) → (𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ↔ (𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑥 ⊆ ℝ)))
22	1, 20, 21	mp2an 691	. . . . 5 ⊢ (𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ↔ (𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑥 ⊆ ℝ))
23	16, 19, 22	sylanbrc 584	. . . 4 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
24	23	ssriv 3986	. . 3 ⊢ (topGen‘ran (,)) ⊆ ((ordTop‘ ≤ ) ↾_t ℝ)
25		eqid 2733	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
26		eqid 2733	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
27		eqid 2733	. . . . . . 7 ⊢ ran (,) = ran (,)
28	25, 26, 27	leordtval 22709	. . . . . 6 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
29	28	oveq1i 7416	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ↾_t ℝ)
30	28, 1	eqeltrri 2831	. . . . . . 7 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
31		tgclb 22465	. . . . . . 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 230	. . . . . 6 ⊢ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ∈ TopBases
33		reex 11198	. . . . . 6 ⊢ ℝ ∈ V
34		tgrest 22655	. . . . . 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 691	. . . . 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 2764	. . . 4 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ))
37		retopbas 24269	. . . . 5 ⊢ ran (,) ∈ TopBases
38		elrest 17370	. . . . . . . 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 691	. . . . . . 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 4148	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↔ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∨ 𝑣 ∈ ran (,)))
41		elun 4148	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ↔ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∨ 𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))))
42		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
43	42	elrnmpt 5954	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞)))
44	43	elv 3481	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞))
45		simpl 484	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ^*)
46		pnfxr 11265	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ +∞ ∈ ℝ^*
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → +∞ ∈ ℝ^*)
48		rexr 11257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
49	48	adantl 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		df-ioc 13326	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
51	50	elixx3g 13334	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝑥(,]+∞) ↔ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
52	51	baib 537	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
53	45, 47, 49, 52	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
54		pnfge 13107	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
55	49, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ +∞)
56	55	biantrud 533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
57		ltpnf 13097	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → 𝑦 < +∞)
58	57	adantl 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 < +∞)
59	58	biantrud 533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∧ 𝑦 < +∞)))
60	53, 56, 59	3bitr2d 307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 < +∞)))
61	60	pm5.32da 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ^* → ((𝑦 ∈ ℝ ∧ 𝑦 ∈ (𝑥(,]+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝑥 < 𝑦 ∧ 𝑦 < +∞))))
62		elin 3964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ (𝑥(,]+∞) ∧ 𝑦 ∈ ℝ))
63	62	biancomi 464	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ (𝑥(,]+∞)))
64		3anass 1096	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞) ↔ (𝑦 ∈ ℝ ∧ (𝑥 < 𝑦 ∧ 𝑦 < +∞)))
65	61, 63, 64	3bitr4g 314	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
66		elioo2 13362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑦 ∈ (𝑥(,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
67	46, 66	mpan2 690	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ (𝑥(,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
68	65, 67	bitr4d 282	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ 𝑦 ∈ (𝑥(,)+∞)))
69	68	eqrdv 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → ((𝑥(,]+∞) ∩ ℝ) = (𝑥(,)+∞))
70		ioorebas 13425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(,)+∞) ∈ ran (,)
71	69, 70	eqeltrdi 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → ((𝑥(,]+∞) ∩ ℝ) ∈ ran (,))
72		ineq1 4205	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) = ((𝑥(,]+∞) ∩ ℝ))
73	72	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (𝑥(,]+∞) → ((𝑣 ∩ ℝ) ∈ ran (,) ↔ ((𝑥(,]+∞) ∩ ℝ) ∈ ran (,)))
74	71, 73	syl5ibrcom 246	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) ∈ ran (,)))
75	74	rexlimiv 3149	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) ∈ ran (,))
76	44, 75	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) → (𝑣 ∩ ℝ) ∈ ran (,))
77		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
78	77	elrnmpt 5954	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥)))
79	78	elv 3481	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥))
80		mnfxr 11268	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ -∞ ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ ∈ ℝ^*)
82		df-ico 13327	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
83	82	elixx3g 13334	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (-∞[,)𝑥) ↔ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
84	83	baib 537	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
85	81, 45, 49, 84	syl3anc 1372	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
86		mnfle 13111	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
87	49, 86	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ ≤ 𝑦)
88	87	biantrurd 534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
89		mnflt 13100	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → -∞ < 𝑦)
90	89	adantl 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ < 𝑦)
91	90	biantrurd 534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 ↔ (-∞ < 𝑦 ∧ 𝑦 < 𝑥)))
92	85, 88, 91	3bitr2d 307	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ < 𝑦 ∧ 𝑦 < 𝑥)))
93	92	pm5.32da 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ^* → ((𝑦 ∈ ℝ ∧ 𝑦 ∈ (-∞[,)𝑥)) ↔ (𝑦 ∈ ℝ ∧ (-∞ < 𝑦 ∧ 𝑦 < 𝑥))))
94		elin 3964	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ (-∞[,)𝑥) ∧ 𝑦 ∈ ℝ))
95	94	biancomi 464	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ (-∞[,)𝑥)))
96		3anass 1096	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ ∧ (-∞ < 𝑦 ∧ 𝑦 < 𝑥)))
97	93, 95, 96	3bitr4g 314	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
98		elioo2 13362	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 ∈ (-∞(,)𝑥) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
99	80, 98	mpan 689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ (-∞(,)𝑥) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
100	97, 99	bitr4d 282	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ^* → (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ 𝑦 ∈ (-∞(,)𝑥)))
101	100	eqrdv 2731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → ((-∞[,)𝑥) ∩ ℝ) = (-∞(,)𝑥))
102		ioorebas 13425	. . . . . . . . . . . . . . . . 17 ⊢ (-∞(,)𝑥) ∈ ran (,)
103	101, 102	eqeltrdi 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → ((-∞[,)𝑥) ∩ ℝ) ∈ ran (,))
104		ineq1 4205	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) = ((-∞[,)𝑥) ∩ ℝ))
105	104	eleq1d 2819	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 = (-∞[,)𝑥) → ((𝑣 ∩ ℝ) ∈ ran (,) ↔ ((-∞[,)𝑥) ∩ ℝ) ∈ ran (,)))
106	103, 105	syl5ibrcom 246	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ^* → (𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) ∈ ran (,)))
107	106	rexlimiv 3149	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) ∈ ran (,))
108	79, 107	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) → (𝑣 ∩ ℝ) ∈ ran (,))
109	76, 108	jaoi 856	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∨ 𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) → (𝑣 ∩ ℝ) ∈ ran (,))
110	41, 109	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) → (𝑣 ∩ ℝ) ∈ ran (,))
111		elssuni 4941	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (,) → 𝑣 ⊆ ∪ ran (,))
112		unirnioo 13423	. . . . . . . . . . . . . 14 ⊢ ℝ = ∪ ran (,)
113	111, 112	sseqtrrdi 4033	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (,) → 𝑣 ⊆ ℝ)
114		df-ss 3965	. . . . . . . . . . . . 13 ⊢ (𝑣 ⊆ ℝ ↔ (𝑣 ∩ ℝ) = 𝑣)
115	113, 114	sylib 217	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran (,) → (𝑣 ∩ ℝ) = 𝑣)
116		id 22	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran (,) → 𝑣 ∈ ran (,))
117	115, 116	eqeltrd 2834	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran (,) → (𝑣 ∩ ℝ) ∈ ran (,))
118	110, 117	jaoi 856	. . . . . . . . . 10 ⊢ ((𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∨ 𝑣 ∈ ran (,)) → (𝑣 ∩ ℝ) ∈ ran (,))
119	40, 118	sylbi 216	. . . . . . . . 9 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) → (𝑣 ∩ ℝ) ∈ ran (,))
120		eleq1 2822	. . . . . . . . 9 ⊢ (𝑢 = (𝑣 ∩ ℝ) → (𝑢 ∈ ran (,) ↔ (𝑣 ∩ ℝ) ∈ ran (,)))
121	119, 120	syl5ibrcom 246	. . . . . . . 8 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) → (𝑢 = (𝑣 ∩ ℝ) → 𝑢 ∈ ran (,)))
122	121	rexlimiv 3149	. . . . . . 7 ⊢ (∃𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))𝑢 = (𝑣 ∩ ℝ) → 𝑢 ∈ ran (,))
123	39, 122	sylbi 216	. . . . . 6 ⊢ (𝑢 ∈ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) → 𝑢 ∈ ran (,))
124	123	ssriv 3986	. . . . 5 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) ⊆ ran (,)
125		tgss 22463	. . . . 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 691	. . . 4 ⊢ (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ)) ⊆ (topGen‘ran (,))
127	36, 126	eqsstri 4016	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) ⊆ (topGen‘ran (,))
128	24, 127	eqssi 3998	. 2 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
129		xrtgioo.1	. 2 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t ℝ)
130	128, 129	eqtr4i 2764	1 ⊢ (topGen‘ran (,)) = 𝐽

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 𝒫 cpw 4602 ∪ cuni 4908 class class class wbr 5148 ↦ cmpt 5231 × cxp 5674 ran crn 5677 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ℝcr 11106 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 (,)cioo 13321 (,]cioc 13322 [,)cico 13323 ↾_t crest 17363 topGenctg 17380 ordTopcordt 17442 Topctop 22387 TopOnctopon 22404 TopBasesctb 22440

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fi 9403 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-rest 17365 df-topgen 17386 df-ordt 17444 df-ps 18516 df-tsr 18517 df-top 22388 df-topon 22405 df-bases 22441

This theorem is referenced by: xrrest 24315 xrsmopn 24320 xrge0tsms 24342 metdcn2 24347 xrge0tsmsd 32197 xrtgcntopre 44176 xrtgioo2 44272

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