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Theorem iocopnst 24857

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 24673	. . . . 5 ⊢ (𝐶(,)(𝐵 + 1)) ∈ (topGen‘ran (,))
2		simp1 1136	. . . . . . . . . . 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 1137	. . . . . . . . . . 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 11953	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 < (𝐵 + 1))
7	6	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝐵 < (𝐵 + 1))
8		peano2re 11278	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ)
9		lelttr 11195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑣 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
10	9	3expa 1118	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑣 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
11	10	ancom1s 653	. . . . . . . . . . . . . . . . 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 689	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
14	7, 13	mpand 695	. . . . . . . . . . . . . 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 1171	. . . . . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
19	3, 5, 18	3jcad 1129	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
20		rexr 11150	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
21		elico2 13302	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
22	20, 21	sylan2 593	. . . . . . . . . . . 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 11195	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣) → 𝐴 < 𝑣))
25		ltle 11193	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 < 𝑣 → 𝐴 ≤ 𝑣))
26	25	3adant2 1131	. . . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . 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 660	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣)) → 𝐴 ≤ 𝑣)
31	30	3adantr3 1172	. . . . . . . . . . . . . . 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 1172	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
35	34	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
36	23, 35	syldan 591	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
37		simp3 1138	. . . . . . . . . . 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 1129	. . . . . . . . 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 1192	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ∈ ℝ)
42		simpl2 1193	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐶 < 𝑣)
43		simpr3 1197	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ≤ 𝐵)
44	41, 42, 43	3jca 1128	. . . . . . . 8 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵))
45	40, 44	impbid1 225	. . . . . . 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 1142	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ)
47	46	rexrd 11154	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ^*)
48		simplr 768	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ)
49		elioc2 13301	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐶(,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)))
50	47, 48, 49	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)))
51		elin 3916	. . . . . . . 8 ⊢ (𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
52	8	rexrd 11154	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ^*)
53	52	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐵 + 1) ∈ ℝ^*)
54		elioo2 13278	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ (𝐵 + 1) ∈ ℝ^*) → (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
55	47, 53, 54	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
56		elicc2 13303	. . . . . . . . . 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 632	. . . . . . . 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 2728	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
62		ineq1 4161	. . . . . 6 ⊢ (𝑣 = (𝐶(,)(𝐵 + 1)) → (𝑣 ∩ (𝐴[,]𝐵)) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
63	62	rspceeqv 3598	. . . . 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 587	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
65		retop 24669	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
66		ovex 7374	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
67		elrest 17323	. . . . 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 692	. . . 4 ⊢ ((𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
69	64, 68	sylibr 234	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
70		iccssre 13321	. . . . 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 2730	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
73		iocopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
74	72, 73	resubmet 24710	. . . 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 2832	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ∃wrex 3054 Vcvv 3434 ∩ cin 3899 ⊆ wss 3900 class class class wbr 5089 × cxp 5612 ran crn 5615 ↾ cres 5616 ∘ ccom 5618 ‘cfv 6477 (class class class)co 7341 ℝcr 10997 1c1 10999 + caddc 11001 ℝ^*cxr 11137 < clt 11138 ≤ cle 11139 − cmin 11336 (,)cioo 13237 (,]cioc 13238 [,)cico 13239 [,]cicc 13240 abscabs 15133 ↾_t crest 17316 topGenctg 17333 MetOpencmopn 21274 Topctop 22801

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 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074 ax-pre-mulgt0 11075 ax-pre-sup 11076

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3344 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-sup 9321 df-inf 9322 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-div 11767 df-nn 12118 df-2 12180 df-3 12181 df-n0 12374 df-z 12461 df-uz 12725 df-q 12839 df-rp 12883 df-xneg 13003 df-xadd 13004 df-xmul 13005 df-ioo 13241 df-ioc 13242 df-ico 13243 df-icc 13244 df-seq 13901 df-exp 13961 df-cj 14998 df-re 14999 df-im 15000 df-sqrt 15134 df-abs 15135 df-rest 17318 df-topgen 17339 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-top 22802 df-topon 22819 df-bases 22854

This theorem is referenced by: (None)

W3C validator