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Theorem iocopnst 23654

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 23480	. . . . 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 11531	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 < (𝐵 + 1))
7	6	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝐵 < (𝐵 + 1))
8		peano2re 10864	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ)
9		lelttr 10782	. . . . . . . . . . . . . . . . . . 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 652	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
12	11	ancomsd 469	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
13	8, 12	mpidan 688	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
14	7, 13	mpand 694	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 ≤ 𝐵 → 𝑣 < (𝐵 + 1)))
15	14	impr 458	. . . . . . . . . . . . 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 416	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
18	17	ad2antlr 726	. . . . . . . . . 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 10738	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
21		elico2 12856	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
22	20, 21	sylan2 595	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
23	22	biimpa 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
24		lelttr 10782	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣) → 𝐴 < 𝑣))
25		ltle 10780	. . . . . . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣)) → 𝐴 ≤ 𝑣)
30	29	an4s 659	. . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
33	32	anasss 470	. . . . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
36	23, 35	syldan 594	. . . . . . . . . 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 516	. . . . . . . 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 228	. . . . . . 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 10742	. . . . . . . 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 12855	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐶(,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)))
50	47, 48, 49	syl2anc 587	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)))
51		elin 3876	. . . . . . . 8 ⊢ (𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
52	8	rexrd 10742	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ^*)
53	52	ad2antlr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐵 + 1) ∈ ℝ^*)
54		elioo2 12833	. . . . . . . . . 10 ⊢ ((𝐶 ∈ ℝ^* ∧ (𝐵 + 1) ∈ ℝ^*) → (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
55	47, 53, 54	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ↔ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1))))
56		elicc2 12857	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
57	56	adantr 484	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
58	55, 57	anbi12d 633	. . . . . . . 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	syl5bb 286	. . . . . . 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 314	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,]𝐵) ↔ 𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵))))
61	60	eqrdv 2756	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
62		ineq1 4111	. . . . . 6 ⊢ (𝑣 = (𝐶(,)(𝐵 + 1)) → (𝑣 ∩ (𝐴[,]𝐵)) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
63	62	rspceeqv 3558	. . . . 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 590	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
65		retop 23476	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
66		ovex 7189	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
67		elrest 16772	. . . . 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 691	. . . 4 ⊢ ((𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
69	64, 68	sylibr 237	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
70		iccssre 12874	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
71	70	adantr 484	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
72		eqid 2758	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
73		iocopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
74	72, 73	resubmet 23516	. . . 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 2855	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ 𝐽)
77	76	ex 416	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3071 Vcvv 3409 ∩ cin 3859 ⊆ wss 3860 class class class wbr 5036 × cxp 5526 ran crn 5529 ↾ cres 5530 ∘ ccom 5532 ‘cfv 6340 (class class class)co 7156 ℝcr 10587 1c1 10589 + caddc 10591 ℝ^*cxr 10725 < clt 10726 ≤ cle 10727 − cmin 10921 (,)cioo 12792 (,]cioc 12793 [,)cico 12794 [,]cicc 12795 abscabs 14654 ↾_t crest 16765 topGenctg 16782 MetOpencmopn 20169 Topctop 21606

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-cnex 10644 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 ax-pre-mulgt0 10665 ax-pre-sup 10666

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-1st 7699 df-2nd 7700 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-er 8305 df-map 8424 df-en 8541 df-dom 8542 df-sdom 8543 df-sup 8952 df-inf 8953 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-sub 10923 df-neg 10924 df-div 11349 df-nn 11688 df-2 11750 df-3 11751 df-n0 11948 df-z 12034 df-uz 12296 df-q 12402 df-rp 12444 df-xneg 12561 df-xadd 12562 df-xmul 12563 df-ioo 12796 df-ioc 12797 df-ico 12798 df-icc 12799 df-seq 13432 df-exp 13493 df-cj 14519 df-re 14520 df-im 14521 df-sqrt 14655 df-abs 14656 df-rest 16767 df-topgen 16788 df-psmet 20171 df-xmet 20172 df-met 20173 df-bl 20174 df-mopn 20175 df-top 21607 df-topon 21624 df-bases 21659

This theorem is referenced by: (None)

W3C validator