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Theorem xrtgioo 24855

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 23254	. . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Top
2		ioof 13445	. . . . . . . . . . 11 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
3		ffn 6686	. . . . . . . . . . 11 ⊢ ((,):(ℝ^* × ℝ^)⟶𝒫 ℝ → (,) Fn (ℝ^ × ℝ^*))
4	2, 3	ax-mp 5	. . . . . . . . . 10 ⊢ (,) Fn (ℝ^* × ℝ^*)
5		iooordt 23265	. . . . . . . . . . 11 ⊢ (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )
6	5	rgen2w 3080	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ ℝ^* ∀𝑦 ∈ ℝ^* (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )
7		ffnov 7517	. . . . . . . . . 10 ⊢ ((,):(ℝ^* × ℝ^)⟶(ordTop‘ ≤ ) ↔ ((,) Fn (ℝ^ × ℝ^) ∧ ∀𝑥 ∈ ℝ^ ∀𝑦 ∈ ℝ^* (𝑥(,)𝑦) ∈ (ordTop‘ ≤ )))
8	4, 6, 7	mpbir2an 721	. . . . . . . . 9 ⊢ (,):(ℝ^* × ℝ^*)⟶(ordTop‘ ≤ )
9		frn 6694	. . . . . . . . 9 ⊢ ((,):(ℝ^* × ℝ^*)⟶(ordTop‘ ≤ ) → ran (,) ⊆ (ordTop‘ ≤ ))
10	8, 9	ax-mp 5	. . . . . . . 8 ⊢ ran (,) ⊆ (ordTop‘ ≤ )
11		tgss 23016	. . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ran (,) ⊆ (ordTop‘ ≤ )) → (topGen‘ran (,)) ⊆ (topGen‘(ordTop‘ ≤ )))
12	1, 10, 11	mp2an 702	. . . . . . 7 ⊢ (topGen‘ran (,)) ⊆ (topGen‘(ordTop‘ ≤ ))
13		tgtop 23021	. . . . . . . 8 ⊢ ((ordTop‘ ≤ ) ∈ Top → (topGen‘(ordTop‘ ≤ )) = (ordTop‘ ≤ ))
14	1, 13	ax-mp 5	. . . . . . 7 ⊢ (topGen‘(ordTop‘ ≤ )) = (ordTop‘ ≤ )
15	12, 14	sseqtri 3982	. . . . . 6 ⊢ (topGen‘ran (,)) ⊆ (ordTop‘ ≤ )
16	15	sseli 3930	. . . . 5 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ∈ (ordTop‘ ≤ ))
17		retopon 24811	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
18		toponss 22975	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ 𝑥 ∈ (topGen‘ran (,))) → 𝑥 ⊆ ℝ)
19	17, 18	mpan 700	. . . . 5 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ⊆ ℝ)
20		reordt 23266	. . . . . 6 ⊢ ℝ ∈ (ordTop‘ ≤ )
21		restopn2 23225	. . . . . 6 ⊢ (((ordTop‘ ≤ ) ∈ Top ∧ ℝ ∈ (ordTop‘ ≤ )) → (𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ↔ (𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑥 ⊆ ℝ)))
22	1, 20, 21	mp2an 702	. . . . 5 ⊢ (𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ) ↔ (𝑥 ∈ (ordTop‘ ≤ ) ∧ 𝑥 ⊆ ℝ))
23	16, 19, 22	sylanbrc 592	. . . 4 ⊢ (𝑥 ∈ (topGen‘ran (,)) → 𝑥 ∈ ((ordTop‘ ≤ ) ↾_t ℝ))
24	23	ssriv 3938	. . 3 ⊢ (topGen‘ran (,)) ⊆ ((ordTop‘ ≤ ) ↾_t ℝ)
25		eqid 2761	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
26		eqid 2761	. . . . . . 7 ⊢ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
27		eqid 2761	. . . . . . 7 ⊢ ran (,) = ran (,)
28	25, 26, 27	leordtval 23261	. . . . . 6 ⊢ (ordTop‘ ≤ ) = (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)))
29	28	oveq1i 7401	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = ((topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ↾_t ℝ)
30	28, 1	eqeltrri 2858	. . . . . . 7 ⊢ (topGen‘((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,))) ∈ Top
31		tgclb 23018	. . . . . . 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 11158	. . . . . 6 ⊢ ℝ ∈ V
34		tgrest 23207	. . . . . 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 702	. . . . 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 2787	. . . 4 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) = (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ))
37		retopbas 24808	. . . . 5 ⊢ ran (,) ∈ TopBases
38		elrest 17447	. . . . . . . 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 702	. . . . . . 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 4104	. . . . . . . . . 10 ⊢ (𝑣 ∈ ((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↔ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∨ 𝑣 ∈ ran (,)))
41		elun 4104	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ (ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ↔ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∨ 𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))))
42		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) = (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞))
43	42	elrnmpt 5930	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞)))
44	43	elv 3458	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞))
45		simpl 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑥 ∈ ℝ^*)
46		pnfxr 11230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ +∞ ∈ ℝ^*
47	46	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → +∞ ∈ ℝ^*)
48		rexr 11222	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℝ^*)
49	48	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ^*)
50		df-ioc 13348	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (,] = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 < 𝑐 ∧ 𝑐 ≤ 𝑏)})
51	50	elixx3g 13356	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (𝑥(,]+∞) ↔ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
52	51	baib 543	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
53	45, 47, 49, 52	syl3anc 1389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝑥(,]+∞) ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
54		pnfge 13126	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ^* → 𝑦 ≤ +∞)
55	49, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 ≤ +∞)
56	55	biantrud 539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑥 < 𝑦 ↔ (𝑥 < 𝑦 ∧ 𝑦 ≤ +∞)))
57		ltpnf 13116	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → 𝑦 < +∞)
58	57	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → 𝑦 < +∞)
59	58	biantrud 539	. . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ^* → ((𝑦 ∈ ℝ ∧ 𝑦 ∈ (𝑥(,]+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝑥 < 𝑦 ∧ 𝑦 < +∞))))
62		elin 3918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ (𝑥(,]+∞) ∧ 𝑦 ∈ ℝ))
63	62	biancomi 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ((𝑥(,]+∞) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ (𝑥(,]+∞)))
64		3anass 1105	. . . . . . . . . . . . . . . . . . . 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 13384	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → (𝑦 ∈ (𝑥(,)+∞) ↔ (𝑦 ∈ ℝ ∧ 𝑥 < 𝑦 ∧ 𝑦 < +∞)))
67	46, 66	mpan2 701	. . . . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → ((𝑥(,]+∞) ∩ ℝ) = (𝑥(,)+∞))
70		ioorebas 13449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥(,)+∞) ∈ ran (,)
71	69, 70	eqeltrdi 2869	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → ((𝑥(,]+∞) ∩ ℝ) ∈ ran (,))
72		ineq1 4163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) = ((𝑥(,]+∞) ∩ ℝ))
73	72	eleq1d 2846	. . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ℝ^* 𝑣 = (𝑥(,]+∞) → (𝑣 ∩ ℝ) ∈ ran (,))
76	44, 75	sylbi 219	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) → (𝑣 ∩ ℝ) ∈ ran (,))
77		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) = (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))
78	77	elrnmpt 5930	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ V → (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥)))
79	78	elv 3458	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥)) ↔ ∃𝑥 ∈ ℝ^* 𝑣 = (-∞[,)𝑥))
80		mnfxr 11233	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ -∞ ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ ∈ ℝ^*)
82		df-ico 13349	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ [,) = (𝑎 ∈ ℝ^, 𝑏 ∈ ℝ^ ↦ {𝑐 ∈ ℝ^* ∣ (𝑎 ≤ 𝑐 ∧ 𝑐 < 𝑏)})
83	82	elixx3g 13356	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (-∞[,)𝑥) ↔ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) ∧ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
84	83	baib 543	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ^*) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
85	81, 45, 49, 84	syl3anc 1389	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (-∞[,)𝑥) ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
86		mnfle 13131	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ^* → -∞ ≤ 𝑦)
87	49, 86	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ ≤ 𝑦)
88	87	biantrurd 540	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → (𝑦 < 𝑥 ↔ (-∞ ≤ 𝑦 ∧ 𝑦 < 𝑥)))
89		mnflt 13119	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℝ → -∞ < 𝑦)
90	89	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ ℝ^* ∧ 𝑦 ∈ ℝ) → -∞ < 𝑦)
91	90	biantrurd 540	. . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ ℝ^* → ((𝑦 ∈ ℝ ∧ 𝑦 ∈ (-∞[,)𝑥)) ↔ (𝑦 ∈ ℝ ∧ (-∞ < 𝑦 ∧ 𝑦 < 𝑥))))
94		elin 3918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ (-∞[,)𝑥) ∧ 𝑦 ∈ ℝ))
95	94	biancomi 466	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ((-∞[,)𝑥) ∩ ℝ) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ (-∞[,)𝑥)))
96		3anass 1105	. . . . . . . . . . . . . . . . . . . 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 13384	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → (𝑦 ∈ (-∞(,)𝑥) ↔ (𝑦 ∈ ℝ ∧ -∞ < 𝑦 ∧ 𝑦 < 𝑥)))
99	80, 98	mpan 700	. . . . . . . . . . . . . . . . . . 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 2759	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ^* → ((-∞[,)𝑥) ∩ ℝ) = (-∞(,)𝑥))
102		ioorebas 13449	. . . . . . . . . . . . . . . . 17 ⊢ (-∞(,)𝑥) ∈ ran (,)
103	101, 102	eqeltrdi 2869	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ^* → ((-∞[,)𝑥) ∩ ℝ) ∈ ran (,))
104		ineq1 4163	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 = (-∞[,)𝑥) → (𝑣 ∩ ℝ) = ((-∞[,)𝑥) ∩ ℝ))
105	104	eleq1d 2846	. . . . . . . . . . . . . . . 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 3155	. . . . . . . . . . . . . 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 868	. . . . . . . . . . . 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 4894	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ ran (,) → 𝑣 ⊆ ∪ ran (,))
112		unirnioo 13447	. . . . . . . . . . . . . 14 ⊢ ℝ = ∪ ran (,)
113	111, 112	sseqtrrdi 3975	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ ran (,) → 𝑣 ⊆ ℝ)
114		dfss2 3920	. . . . . . . . . . . . 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 2861	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran (,) → (𝑣 ∩ ℝ) ∈ ran (,))
118	110, 117	jaoi 868	. . . . . . . . . 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 2849	. . . . . . . . 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 3155	. . . . . . 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 3938	. . . . 5 ⊢ (((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ) ⊆ ran (,)
125		tgss 23016	. . . . 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 702	. . . 4 ⊢ (topGen‘(((ran (𝑥 ∈ ℝ^* ↦ (𝑥(,]+∞)) ∪ ran (𝑥 ∈ ℝ^* ↦ (-∞[,)𝑥))) ∪ ran (,)) ↾_t ℝ)) ⊆ (topGen‘ran (,))
127	36, 126	eqsstri 3980	. . 3 ⊢ ((ordTop‘ ≤ ) ↾_t ℝ) ⊆ (topGen‘ran (,))
128	24, 127	eqssi 3950	. 2 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾_t ℝ)
129		xrtgioo.1	. 2 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾_t ℝ)
130	128, 129	eqtr4i 2787	1 ⊢ (topGen‘ran (,)) = 𝐽

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 Vcvv 3453 ∪ cun 3900 ∩ cin 3901 ⊆ wss 3902 𝒫 cpw 4552 ∪ cuni 4862 class class class wbr 5097 ↦ cmpt 5178 × cxp 5641 ran crn 5644 Fn wfn 6511 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ℝcr 11066 +∞cpnf 11207 -∞cmnf 11208 ℝ^*cxr 11209 < clt 11210 ≤ cle 11211 (,)cioo 13343 (,]cioc 13344 [,)cico 13345 ↾_t crest 17440 topGenctg 17457 ordTopcordt 17520 Topctop 22941 TopOnctopon 22958 TopBasesctb 22993

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9351 df-sup 9382 df-inf 9383 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-n0 12476 df-z 12563 df-uz 12834 df-q 12944 df-ioo 13347 df-ioc 13348 df-ico 13349 df-icc 13350 df-rest 17442 df-topgen 17463 df-ordt 17522 df-ps 18589 df-tsr 18590 df-top 22942 df-topon 22959 df-bases 22994

This theorem is referenced by: xrrest 24856 xrsmopn 24861 xrge0tsms 24883 metdcn2 24888 xrge0tsmsd 33214 xrtgcntopre 46013 xrtgioo2 46107

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