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Theorem icoopnst 24969

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 24786	. . . . 5 ⊢ ((𝐴 − 1)(,)𝐶) ∈ (topGen‘ran (,))
2		simp1 1137	. . . . . . . . . . 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 12109	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) < 𝐴)
5	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 − 1) < 𝐴)
6		peano2rem 11576	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
7	6	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 − 1) ∈ ℝ)
8		ltletr 11353	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
9	8	3expb 1121	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 − 1) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ)) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
10	7, 9	mpancom 688	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (((𝐴 − 1) < 𝐴 ∧ 𝐴 ≤ 𝑣) → (𝐴 − 1) < 𝑣))
11	5, 10	mpand 695	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝐴 ≤ 𝑣 → (𝐴 − 1) < 𝑣))
12	11	impr 454	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣)) → (𝐴 − 1) < 𝑣)
13	12	3adantr3 1172	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)) → (𝐴 − 1) < 𝑣)
14	13	ex 412	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝐴 − 1) < 𝑣))
15	14	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝐴 − 1) < 𝑣))
16		simp3 1139	. . . . . . . . . . 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 1130	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
19		simp2 1138	. . . . . . . . . . 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 11307	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		elioc2 13450	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
23	21, 22	sylan 580	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
24	23	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵))
25		ltleletr 11354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑣 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
26	25	3expa 1119	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
27	26	an31s 654	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) → ((𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝑣 ≤ 𝐵))
28	27	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝑣 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → 𝑣 ≤ 𝐵)
29	28	ancom2s 650	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝐶 ≤ 𝐵 ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
30	29	an4s 660	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) ∧ (𝑣 ∈ ℝ ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
31	30	3adantr2 1171	. . . . . . . . . . . . . . 15 ⊢ ((((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)) → 𝑣 ≤ 𝐵)
32	31	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐶 ≤ 𝐵) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
33	32	anasss 466	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
34	33	3adantr2 1171	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
35	34	adantll 714	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐶 ∈ ℝ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
36	24, 35	syldan 591	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → 𝑣 ≤ 𝐵))
37	3, 20, 36	3jcad 1130	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶) → (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
38	18, 37	jcad 512	. . . . . . . 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 1192	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ∈ ℝ)
40		simpr2 1196	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐴 ≤ 𝑣)
41		simpl3 1194	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 < 𝐶)
42	39, 40, 41	3jca 1129	. . . . . . . 8 ⊢ (((𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶))
43	38, 42	impbid1 225	. . . . . . 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 767	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐴 ∈ ℝ)
45	24	simp1d 1143	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ)
46	45	rexrd 11311	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ∈ ℝ^*)
47		elico2 13451	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ^*) → (𝑣 ∈ (𝐴[,)𝐶) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)))
48	44, 46, 47	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,)𝐶) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 < 𝐶)))
49		elin 3967	. . . . . . . 8 ⊢ (𝑣 ∈ (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
50	6	rexrd 11311	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ^*)
51	50	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴 − 1) ∈ ℝ^*)
52		elioo2 13428	. . . . . . . . . 10 ⊢ (((𝐴 − 1) ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ↔ (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
53	51, 46, 52	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ ((𝐴 − 1)(,)𝐶) ↔ (𝑣 ∈ ℝ ∧ (𝐴 − 1) < 𝑣 ∧ 𝑣 < 𝐶)))
54		elicc2 13452	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
55	54	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
56	53, 55	anbi12d 632	. . . . . . . 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 283	. . . . . . 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 311	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝑣 ∈ (𝐴[,)𝐶) ↔ 𝑣 ∈ (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵))))
59	58	eqrdv 2735	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)))
60		ineq1 4213	. . . . . 6 ⊢ (𝑣 = ((𝐴 − 1)(,)𝐶) → (𝑣 ∩ (𝐴[,]𝐵)) = (((𝐴 − 1)(,)𝐶) ∩ (𝐴[,]𝐵)))
61	60	rspceeqv 3645	. . . . 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 587	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵)))
63		retop 24782	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
64		ovex 7464	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
65		elrest 17472	. . . . 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 692	. . . 4 ⊢ ((𝐴[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐴[,)𝐶) = (𝑣 ∩ (𝐴[,]𝐵)))
67	62, 66	sylibr 234	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
68		iccssre 13469	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
69	68	adantr 480	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
70		eqid 2737	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
71		icoopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
72	70, 71	resubmet 24823	. . . 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 2844	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴(,]𝐵)) → (𝐴[,)𝐶) ∈ 𝐽)
75	74	ex 412	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴(,]𝐵) → (𝐴[,)𝐶) ∈ 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 1c1 11156 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 (,)cioo 13387 (,]cioc 13388 [,)cico 13389 [,]cicc 13390 abscabs 15273 ↾_t crest 17465 topGenctg 17482 MetOpencmopn 21354 Topctop 22899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953

This theorem is referenced by: (None)

W3C validator