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Theorem icoopnst 25253

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 25077	. . . . 5 ⊢ ((𝐴 − 1)(,)𝐶) ∈ (topGen‘ran (,))
2		simp1 1154	. . . . . . . . . . 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 12152	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
5	4	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 − 1) < 𝐴)
6		peano2rem 11618	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
7	6	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 − 1) ∈ ℝ)
8		ltletr 11395	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
9	8	3expb 1138	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 − 1) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
10	7, 9	mpancom 701	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
11	5, 10	mpand 708	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 ≤ 𝑣 → (𝐴 − 1) < 𝑣))
12	11	impr 460	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣)) → (𝐴 − 1) < 𝑣)
13	12	3adantr3 1190	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)) → (𝐴 − 1) < 𝑣)
14	13	ex 418	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝐴 − 1) < 𝑣))
15	14	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝐴 − 1) < 𝑣))
16		simp3 1156	. . . . . . . . . . 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 1147	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
19		simp2 1155	. . . . . . . . . . 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 11348	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		elioc2 13533	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
23	21, 22	sylan 592	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
24	23	biimpa 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
25		ltleletr 11396	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
26	25	3expa 1136	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
27	26	an31s 667	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
28	27	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝑣 ≤ 𝐵)
29	28	ancom2s 663	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝐶 ≤ 𝐵 ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
30	29	an4s 673	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) ∧ (𝑣 ∈ ℝ ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
31	30	3adantr2 1189	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
32	31	ex 418	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
33	32	anasss 472	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
34	33	3adantr2 1189	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
35	34	adantll 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
36	24, 35	syldan 603	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
37	3, 20, 36	3jcad 1147	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
38	18, 37	jcad 522	. . . . . . . 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 1210	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ∈ ℝ)
40		simpr2 1214	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐴 ≤ 𝑣)
41		simpl3 1212	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 < 𝐶)
42	39, 40, 41	3jca 1146	. . . . . . . 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 779	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ)
45	24	simp1d 1160	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)
46	45	rexrd 11352	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ^*)
47		elico2 13534	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ^*) → (𝑣 ∈ (𝐴[,)𝐶) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)))
48	44, 46, 47	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,)𝐶) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)))
49		elin 3915	. . . . . . . 8 ⊢ (𝑣 ∈ (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
50	6	rexrd 11352	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ^*)
51	50	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴 − 1) ∈ ℝ^*)
52		elioo2 13510	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ↔ (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
53	51, 46, 52	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ↔ (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
54		elicc2 13535	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
55	54	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
56	53, 55	anbi12d 644	. . . . . . . 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	bitrid 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 2759	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)))
60		ineq1 4159	. . . . . 6 ⊢ (𝑣 = ((𝐴 − 1)(,)𝐶) → (𝑣 ∩ (𝐴[,]𝐵)) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)))
61	60	rspceeqv 3599	. . . . 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 599	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵)))
63		retop 25073	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
64		ovex 7451	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
65		elrest 17591	. . . . 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 705	. . . 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 13553	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
69	68	adantr 486	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
70		eqid 2761	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
71		icoopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
72	70, 71	resubmet 25114	. . . 4 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
73	69, 72	syl 18	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐽 = ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
74	67, 73	eleqtrrd 2864	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) ∈ 𝐽)
75	74	ex 418	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3087 Vcvv 3451 ∩ cin 3898 ⊆ wss 3899 class class class wbr 5103 × cxp 5649 ran crn 5652 ↾ cres 5653 ∘ ccom 5655 ‘cfv 6537 (class class class)co 7418 ℝcr 11192 1c1 11194 ℝ^*cxr 11335 < clt 11336 ≤ cle 11337 − cmin 11534 (,)cioo 13469 (,]cioc 13470 [,)cico 13471 [,]cicc 13472 abscabs 15394 ↾_t crest 17584 topGenctg 17601 MetOpencmopn 21661 Topctop 23204

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9427 df-inf 9428 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-z 12687 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-ioo 13473 df-ioc 13474 df-ico 13475 df-icc 13476 df-seq 14138 df-exp 14198 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-rest 17586 df-topgen 17607 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-top 23205 df-topon 23222 df-bases 23257

This theorem is used by: (None)

W3C validator