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Theorem icoopnst 23544

Description: A half-open interval starting at 𝐴 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
icoopnst.1	⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))

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

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

Step	Hyp	Ref	Expression
1		iooretop 23371	. . . . 5 ⊢ ((𝐴 − 1)(,)𝐶) ∈ (topGen‘ran (,))
2		simp1 1133	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ∈ ℝ)
3	2	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ∈ ℝ))
4		ltm1 11471	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
5	4	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 − 1) < 𝐴)
6		peano2rem 10942	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
7	6	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 − 1) ∈ ℝ)
8		ltletr 10721	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
9	8	3expb 1117	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 − 1) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
10	7, 9	mpancom 687	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
11	5, 10	mpand 694	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 ≤ 𝑣 → (𝐴 − 1) < 𝑣))
12	11	impr 458	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣)) → (𝐴 − 1) < 𝑣)
13	12	3adantr3 1168	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)) → (𝐴 − 1) < 𝑣)
14	13	ex 416	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝐴 − 1) < 𝑣))
15	14	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝐴 − 1) < 𝑣))
16		simp3 1135	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 < 𝐶)
17	16	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 < 𝐶))
18	3, 15, 17	3jcad 1126	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
19		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝐴 ≤ 𝑣)
20	19	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝐴 ≤ 𝑣))
21		rexr 10676	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		elioc2 12788	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
23	21, 22	sylan 583	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
24	23	biimpa 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
25		ltleletr 10722	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
26	25	3expa 1115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
27	26	an31s 653	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
28	27	imp 410	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝑣 ≤ 𝐵)
29	28	ancom2s 649	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝐶 ≤ 𝐵 ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
30	29	an4s 659	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) ∧ (𝑣 ∈ ℝ ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
31	30	3adantr2 1167	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
32	31	ex 416	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
33	32	anasss 470	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
34	33	3adantr2 1167	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
35	34	adantll 713	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
36	24, 35	syldan 594	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
37	3, 20, 36	3jcad 1126	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
38	18, 37	jcad 516	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → ((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
39		simpl1 1188	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ∈ ℝ)
40		simpr2 1192	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐴 ≤ 𝑣)
41		simpl3 1190	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 < 𝐶)
42	39, 40, 41	3jca 1125	. . . . . . . 8 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶))
43	38, 42	impbid1 228	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) ↔ ((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
44		simpll 766	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ)
45	24	simp1d 1139	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)
46	45	rexrd 10680	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ^*)
47		elico2 12789	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ^*) → (𝑣 ∈ (𝐴[,)𝐶) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)))
48	44, 46, 47	syl2anc 587	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,)𝐶) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)))
49		elin 3897	. . . . . . . 8 ⊢ (𝑣 ∈ (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
50	6	rexrd 10680	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ^*)
51	50	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴 − 1) ∈ ℝ^*)
52		elioo2 12767	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ↔ (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
53	51, 46, 52	syl2anc 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ↔ (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
54		elicc2 12790	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
55	54	adantr 484	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
56	53, 55	anbi12d 633	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ((𝐴 − 1)(,)𝐶) ∧ 𝑣 ∈ (𝐴[,]𝐵)) ↔ ((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
57	49, 56	syl5bb 286	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)) ↔ ((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵))))
58	43, 48, 57	3bitr4d 314	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,)𝐶) ↔ 𝑣 ∈ (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵))))
59	58	eqrdv 2796	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)))
60		ineq1 4131	. . . . . 6 ⊢ (𝑣 = ((𝐴 − 1)(,)𝐶) → (𝑣 ∩ (𝐴[,]𝐵)) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)))
61	60	rspceeqv 3586	. . . . 5 ⊢ ((((𝐴 − 1)(,)𝐶) ∈ (topGen‘ran (,)) ∧ (𝐴[,)𝐶) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵))) → ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵)))
62	1, 59, 61	sylancr 590	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵)))
63		retop 23367	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
64		ovex 7168	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
65		elrest 16693	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ (𝐴[,]𝐵) ∈ V) → ((𝐴[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵))))
66	63, 64, 65	mp2an 691	. . . 4 ⊢ ((𝐴[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵)))
67	62, 66	sylibr 237	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
68		iccssre 12807	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
69	68	adantr 484	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
70		eqid 2798	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
71		icoopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
72	70, 71	resubmet 23407	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
73	69, 72	syl 17	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
74	67, 73	eleqtrrd 2893	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) ∈ 𝐽)
75	74	ex 416	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 1c1 10527 ℝ^*cxr 10663 < clt 10664 ≤ cle 10665 − cmin 10859 (,)cioo 12726 (,]cioc 12727 [,)cico 12728 [,]cicc 12729 abscabs 14585 ↾_t crest 16686 topGenctg 16703 MetOpencmopn 20081 Topctop 21498

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 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551

This theorem is referenced by: (None)

W3C validator