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Theorem iocopnst 24819

Description: A half-open interval ending at 𝐵 is open in the closed interval from 𝐴 to 𝐵. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)

**Hypothesis**
Ref	Expression
iocopnst.1	⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))

**Assertion**
Ref	Expression
iocopnst	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))

Proof of Theorem iocopnst Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iooretop 24637	. . . . 5 ⊢ (𝐶(,)(𝐵 + 1)) ∈ (topGen‘ran (,))
2		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 ∈ ℝ)
3	2	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 ∈ ℝ))
4		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐶 < 𝑣)
5	4	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐶 < 𝑣))
6		ltp1 12058	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 < (𝐵 + 1))
7	6	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝐵 < (𝐵 + 1))
8		peano2re 11391	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ)
9		lelttr 11308	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
10	9	3expa 1115	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
11	10	ancom1s 650	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
12	11	ancomsd 465	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
13	8, 12	mpidan 686	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
14	7, 13	mpand 692	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 ≤ 𝐵 → 𝑣 < (𝐵 + 1)))
15	14	impr 454	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝑣 ≤ 𝐵)) → 𝑣 < (𝐵 + 1))
16	15	3adantr2 1167	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 < (𝐵 + 1))
17	16	ex 412	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
18	17	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
19	3, 5, 18	3jcad 1126	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
20		rexr 11264	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
21		elico2 13394	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
22	20, 21	sylan2 592	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
23	22	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
24		lelttr 11308	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣) → 𝐴 < 𝑣))
25		ltle 11306	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 < 𝑣 → 𝐴 ≤ 𝑣))
26	25	3adant2 1128	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 < 𝑣 → 𝐴 ≤ 𝑣))
27	24, 26	syld 47	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣) → 𝐴 ≤ 𝑣))
28	27	3expa 1115	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) → ((𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣) → 𝐴 ≤ 𝑣))
29	28	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣)) → 𝐴 ≤ 𝑣)
30	29	an4s 657	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣)) → 𝐴 ≤ 𝑣)
31	30	3adantr3 1168	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐴 ≤ 𝑣)
32	31	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
33	32	anasss 466	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
34	33	3adantr3 1168	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
35	34	adantlr 712	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
36	23, 35	syldan 590	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
37		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 ≤ 𝐵)
38	37	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 ≤ 𝐵))
39	3, 36, 38	3jcad 1126	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
40	19, 39	jcad 512	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
41		simpl1 1188	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ∈ ℝ)
42		simpl2 1189	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐶 < 𝑣)
43		simpr3 1193	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ≤ 𝐵)
44	41, 42, 43	3jca 1125	. . . . . . . 8 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵))
45	40, 44	impbid1 224	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) ↔ ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
46	23	simp1d 1139	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ)
47	46	rexrd 11268	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ^*)
48		simplr 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ)
49		elioc2 13393	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐶(,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)))
50	47, 48, 49	syl2anc 583	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)))
51		elin 3959	. . . . . . . 8 ⊢ (𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
52	8	rexrd 11268	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ^*)
53	52	ad2antlr 724	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐵 + 1) ∈ ℝ^*)
54		elioo2 13371	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ (𝐵 + 1) ∈ ℝ^*) → (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
55	47, 53, 54	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
56		elicc2 13395	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
57	56	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
58	55, 57	anbi12d 630	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ (𝐶(,)(𝐵 + 1)) ∧ 𝑣 ∈ (𝐴[,]𝐵)) ↔ ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
59	51, 58	bitrid 283	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)) ↔ ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
60	45, 50, 59	3bitr4d 311	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,]𝐵) ↔ 𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵))))
61	60	eqrdv 2724	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
62		ineq1 4200	. . . . . 6 ⊢ (𝑣 = (𝐶(,)(𝐵 + 1)) → (𝑣 ∩ (𝐴[,]𝐵)) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
63	62	rspceeqv 3628	. . . . 5 ⊢ (((𝐶(,)(𝐵 + 1)) ∈ (topGen‘ran (,)) ∧ (𝐶(,]𝐵) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵))) → ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
64	1, 61, 63	sylancr 586	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
65		retop 24633	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
66		ovex 7438	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
67		elrest 17382	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ∈ V) → ((𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵))))
68	65, 66, 67	mp2an 689	. . . 4 ⊢ ((𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
69	64, 68	sylibr 233	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
70		iccssre 13412	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
71	70	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
72		eqid 2726	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
73		iocopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
74	72, 73	resubmet 24673	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
75	71, 74	syl 17	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
76	69, 75	eleqtrrd 2830	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ 𝐽)
77	76	ex 412	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 class class class wbr 5141 × cxp 5667 ran crn 5670 ↾ cres 5671 ∘ ccom 5673 ‘cfv 6537 (class class class)co 7405 ℝcr 11111 1c1 11113 + caddc 11115 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 − cmin 11448 (,)cioo 13330 (,]cioc 13331 [,)cico 13332 [,]cicc 13333 abscabs 15187 ↾_t crest 17375 topGenctg 17392 MetOpencmopn 21230 Topctop 22750

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-rest 17377 df-topgen 17398 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-top 22751 df-topon 22768 df-bases 22804

This theorem is referenced by: (None)

W3C validator