iocopnst - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > iocopnst		Structured version Visualization version GIF version

Theorem iocopnst 24385

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 24211	. . . . 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 12036	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → 𝐵 < (𝐵 + 1))
7	6	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → 𝐵 < (𝐵 + 1))
8		peano2re 11369	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ)
9		lelttr 11286	. . . . . . . . . . . . . . . . . . 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 651	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝑣 ≤ 𝐵 ∧ 𝐵 < (𝐵 + 1)) → 𝑣 < (𝐵 + 1)))
12	11	ancomsd 466	. . . . . . . . . . . . . . . 16 ⊢ (((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) ∧ (𝐵 + 1) ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
13	8, 12	mpidan 687	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐵 < (𝐵 + 1) ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
14	7, 13	mpand 693	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 ≤ 𝐵 → 𝑣 < (𝐵 + 1)))
15	14	impr 455	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝑣 ≤ 𝐵)) → 𝑣 < (𝐵 + 1))
16	15	3adantr2 1170	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 < (𝐵 + 1))
17	16	ex 413	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝑣 < (𝐵 + 1)))
18	17	ad2antlr 725	. . . . . . . . . 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 11242	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
21		elico2 13370	. . . . . . . . . . . . 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 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
24		lelttr 11286	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣) → 𝐴 < 𝑣))
25		ltle 11284	. . . . . . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝑣 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝑣)) → 𝐴 ≤ 𝑣)
30	29	an4s 658	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣)) → 𝐴 ≤ 𝑣)
31	30	3adantr3 1171	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) ∧ (𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐴 ≤ 𝑣)
32	31	ex 413	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) ∧ 𝐴 ≤ 𝐶) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
33	32	anasss 467	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
34	33	3adantr3 1171	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ ∧ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → ((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 ≤ 𝐵) → 𝐴 ≤ 𝑣))
35	34	adantlr 713	. . . . . . . . . . 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 513	. . . . . . . 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 1191	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝑣 ∈ ℝ)
42		simpl2 1192	. . . . . . . . 9 ⊢ (((𝑣 ∈ ℝ ∧ 𝐶 < 𝑣 ∧ 𝑣 < (𝐵 + 1)) ∧ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)) → 𝐶 < 𝑣)
43		simpr3 1196	. . . . . . . . 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 224	. . . . . . 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 11246	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐶 ∈ ℝ^*)
48		simplr 767	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ)
49		elioc2 13369	. . . . . . . 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 3960	. . . . . . . 8 ⊢ (𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)) ↔ (𝑣 ∈ (𝐶(,)(𝐵 + 1)) ∧ 𝑣 ∈ (𝐴[,]𝐵)))
52	8	rexrd 11246	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → (𝐵 + 1) ∈ ℝ^*)
53	52	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐵 + 1) ∈ ℝ^*)
54		elioo2 13347	. . . . . . . . . 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 13371	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
57	56	adantr 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐴[,]𝐵) ↔ (𝑣 ∈ ℝ ∧ 𝐴 ≤ 𝑣 ∧ 𝑣 ≤ 𝐵)))
58	55, 57	anbi12d 631	. . . . . . . 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 282	. . . . . . 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 310	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝑣 ∈ (𝐶(,]𝐵) ↔ 𝑣 ∈ ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵))))
61	60	eqrdv 2729	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
62		ineq1 4201	. . . . . 6 ⊢ (𝑣 = (𝐶(,)(𝐵 + 1)) → (𝑣 ∩ (𝐴[,]𝐵)) = ((𝐶(,)(𝐵 + 1)) ∩ (𝐴[,]𝐵)))
63	62	rspceeqv 3629	. . . . 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 24207	. . . . 5 ⊢ (topGen‘ran (,)) ∈ Top
66		ovex 7426	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ V
67		elrest 17355	. . . . 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 690	. . . 4 ⊢ ((𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)) ↔ ∃𝑣 ∈ (topGen‘ran (,))(𝐶(,]𝐵) = (𝑣 ∩ (𝐴[,]𝐵)))
69	64, 68	sylibr 233	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ ((topGen‘ran (,)) ↾_t (𝐴[,]𝐵)))
70		iccssre 13388	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
71	70	adantr 481	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐴[,]𝐵) ⊆ ℝ)
72		eqid 2731	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
73		iocopnst.1	. . . . 5 ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ ((𝐴[,]𝐵) × (𝐴[,]𝐵))))
74	72, 73	resubmet 24247	. . . 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 2835	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐶 ∈ (𝐴[,)𝐵)) → (𝐶(,]𝐵) ∈ 𝐽)
77	76	ex 413	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 ∈ (𝐴[,)𝐵) → (𝐶(,]𝐵) ∈ 𝐽))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3069 Vcvv 3473 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5141 × cxp 5667 ran crn 5670 ↾ cres 5671 ∘ ccom 5673 ‘cfv 6532 (class class class)co 7393 ℝcr 11091 1c1 11093 + caddc 11095 ℝ^*cxr 11229 < clt 11230 ≤ cle 11231 − cmin 11426 (,)cioo 13306 (,]cioc 13307 [,)cico 13308 [,]cicc 13309 abscabs 15163 ↾_t crest 17348 topGenctg 17365 MetOpencmopn 20868 Topctop 22324

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-n0 12455 df-z 12541 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-rest 17350 df-topgen 17371 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-top 22325 df-topon 22342 df-bases 22378

This theorem is referenced by: (None)

W3C validator