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Theorem ismbf3d 25163

Description: Simplified form of ismbfd 25148. (Contributed by Mario Carneiro, 18-Jun-2014.)

**Hypotheses**
Ref	Expression
ismbf3d.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
ismbf3d.2	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)

**Assertion**
Ref	Expression
ismbf3d	⊢ (𝜑 → 𝐹 ∈ MblFn)

Distinct variable groups: 𝑥,𝐹 𝜑,𝑥 Allowed substitution hint: 𝐴(𝑥)

Proof of Theorem ismbf3d Dummy variables 𝑣 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismbf3d.1	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
2		fimacnv 6737	. . . 4 ⊢ (𝐹:𝐴⟶ℝ → (^◡𝐹 “ ℝ) = 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → (^◡𝐹 “ ℝ) = 𝐴)
4		imaiun 7241	. . . . 5 ⊢ (^◡𝐹 “ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞)) = ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-𝑦(,)+∞))
5		ioossre 13382	. . . . . . . . 9 ⊢ (-𝑦(,)+∞) ⊆ ℝ
6	5	rgenw 3066	. . . . . . . 8 ⊢ ∀𝑦 ∈ ℕ (-𝑦(,)+∞) ⊆ ℝ
7		iunss 5048	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ ℕ (-𝑦(,)+∞) ⊆ ℝ ↔ ∀𝑦 ∈ ℕ (-𝑦(,)+∞) ⊆ ℝ)
8	6, 7	mpbir 230	. . . . . . 7 ⊢ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞) ⊆ ℝ
9		renegcl 11520	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ → -𝑧 ∈ ℝ)
10		arch 12466	. . . . . . . . . . 11 ⊢ (-𝑧 ∈ ℝ → ∃𝑦 ∈ ℕ -𝑧 < 𝑦)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → ∃𝑦 ∈ ℕ -𝑧 < 𝑦)
12		simpl 484	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → 𝑧 ∈ ℝ)
13	12	biantrurd 534	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → (-𝑦 < 𝑧 ↔ (𝑧 ∈ ℝ ∧ -𝑦 < 𝑧)))
14		nnre 12216	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
15		ltnegcon1 11712	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑧 < 𝑦 ↔ -𝑦 < 𝑧))
16	14, 15	sylan2 594	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → (-𝑧 < 𝑦 ↔ -𝑦 < 𝑧))
17	14	adantl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℝ)
18	17	renegcld 11638	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → -𝑦 ∈ ℝ)
19	18	rexrd 11261	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → -𝑦 ∈ ℝ^*)
20		elioopnf 13417	. . . . . . . . . . . . 13 ⊢ (-𝑦 ∈ ℝ^* → (𝑧 ∈ (-𝑦(,)+∞) ↔ (𝑧 ∈ ℝ ∧ -𝑦 < 𝑧)))
21	19, 20	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → (𝑧 ∈ (-𝑦(,)+∞) ↔ (𝑧 ∈ ℝ ∧ -𝑦 < 𝑧)))
22	13, 16, 21	3bitr4d 311	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℝ ∧ 𝑦 ∈ ℕ) → (-𝑧 < 𝑦 ↔ 𝑧 ∈ (-𝑦(,)+∞)))
23	22	rexbidva 3177	. . . . . . . . . 10 ⊢ (𝑧 ∈ ℝ → (∃𝑦 ∈ ℕ -𝑧 < 𝑦 ↔ ∃𝑦 ∈ ℕ 𝑧 ∈ (-𝑦(,)+∞)))
24	11, 23	mpbid 231	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ → ∃𝑦 ∈ ℕ 𝑧 ∈ (-𝑦(,)+∞))
25		eliun 5001	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞) ↔ ∃𝑦 ∈ ℕ 𝑧 ∈ (-𝑦(,)+∞))
26	24, 25	sylibr 233	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞))
27	26	ssriv 3986	. . . . . . 7 ⊢ ℝ ⊆ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞)
28	8, 27	eqssi 3998	. . . . . 6 ⊢ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞) = ℝ
29	28	imaeq2i 6056	. . . . 5 ⊢ (^◡𝐹 “ ∪ 𝑦 ∈ ℕ (-𝑦(,)+∞)) = (^◡𝐹 “ ℝ)
30	4, 29	eqtr3i 2763	. . . 4 ⊢ ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-𝑦(,)+∞)) = (^◡𝐹 “ ℝ)
31		ismbf3d.2	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
32	31	ralrimiva 3147	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
33	14	renegcld 11638	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → -𝑦 ∈ ℝ)
34		oveq1 7413	. . . . . . . . . 10 ⊢ (𝑥 = -𝑦 → (𝑥(,)+∞) = (-𝑦(,)+∞))
35	34	imaeq2d 6058	. . . . . . . . 9 ⊢ (𝑥 = -𝑦 → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ (-𝑦(,)+∞)))
36	35	eleq1d 2819	. . . . . . . 8 ⊢ (𝑥 = -𝑦 → ((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ↔ (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol))
37	36	rspccva 3612	. . . . . . 7 ⊢ ((∀𝑥 ∈ ℝ (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ∧ -𝑦 ∈ ℝ) → (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol)
38	32, 33, 37	syl2an 597	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℕ) → (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol)
39	38	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ ℕ (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol)
40		iunmbl 25062	. . . . 5 ⊢ (∀𝑦 ∈ ℕ (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol → ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol)
41	39, 40	syl 17	. . . 4 ⊢ (𝜑 → ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-𝑦(,)+∞)) ∈ dom vol)
42	30, 41	eqeltrrid 2839	. . 3 ⊢ (𝜑 → (^◡𝐹 “ ℝ) ∈ dom vol)
43	3, 42	eqeltrrd 2835	. 2 ⊢ (𝜑 → 𝐴 ∈ dom vol)
44		imaiun 7241	. . . . . . 7 ⊢ (^◡𝐹 “ ∪ 𝑦 ∈ ℕ (-∞(,](𝑧 − (1 / 𝑦)))) = ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦))))
45		eliun 5001	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ ℕ (-∞(,](𝑧 − (1 / 𝑦))) ↔ ∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))))
46		3simpb 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦))) → (𝑥 ∈ ℝ ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦))))
47		simplr 768	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → 𝑧 ∈ ℝ)
48		nnrp 12982	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ⁺)
49	48	ad2antrl 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → 𝑦 ∈ ℝ⁺)
50	49	rpreccld 13023	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → (1 / 𝑦) ∈ ℝ⁺)
51	47, 50	ltsubrpd 13045	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → (𝑧 − (1 / 𝑦)) < 𝑧)
52		simprr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → 𝑥 ∈ ℝ)
53		simpr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ)
54		nnrecre 12251	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ ℕ → (1 / 𝑦) ∈ ℝ)
55		resubcl 11521	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ ℝ ∧ (1 / 𝑦) ∈ ℝ) → (𝑧 − (1 / 𝑦)) ∈ ℝ)
56	53, 54, 55	syl2an 597	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (𝑧 − (1 / 𝑦)) ∈ ℝ)
57	56	adantrr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → (𝑧 − (1 / 𝑦)) ∈ ℝ)
58		lelttr 11301	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ ℝ ∧ (𝑧 − (1 / 𝑦)) ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 ≤ (𝑧 − (1 / 𝑦)) ∧ (𝑧 − (1 / 𝑦)) < 𝑧) → 𝑥 < 𝑧))
59	52, 57, 47, 58	syl3anc 1372	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → ((𝑥 ≤ (𝑧 − (1 / 𝑦)) ∧ (𝑧 − (1 / 𝑦)) < 𝑧) → 𝑥 < 𝑧))
60	51, 59	mpan2d 693	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑦 ∈ ℕ ∧ 𝑥 ∈ ℝ)) → (𝑥 ≤ (𝑧 − (1 / 𝑦)) → 𝑥 < 𝑧))
61	60	anassrs 469	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑥 ≤ (𝑧 − (1 / 𝑦)) → 𝑥 < 𝑧))
62	61	imdistanda 573	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((𝑥 ∈ ℝ ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦))) → (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
63	46, 62	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦))) → (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
64		mnfxr 11268	. . . . . . . . . . . . . . . 16 ⊢ -∞ ∈ ℝ^*
65		elioc2 13384	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ (𝑧 − (1 / 𝑦)) ∈ ℝ) → (𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦)))))
66	64, 56, 65	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦)))))
67		rexr 11257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ ℝ → 𝑧 ∈ ℝ^*)
68	67	adantl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → 𝑧 ∈ ℝ^*)
69		elioomnf 13418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ℝ^* → (𝑥 ∈ (-∞(,)𝑧) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
70	68, 69	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑥 ∈ (-∞(,)𝑧) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
71	70	adantr 482	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (𝑥 ∈ (-∞(,)𝑧) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
72	63, 66, 71	3imtr4d 294	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) → 𝑥 ∈ (-∞(,)𝑧)))
73	72	rexlimdva 3156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) → 𝑥 ∈ (-∞(,)𝑧)))
74	73, 70	sylibd 238	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) → (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
75		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → 𝑥 ∈ ℝ)
76	75	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → 𝑥 ∈ ℝ)
77	76	mnfltd 13101	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → -∞ < 𝑥)
78	56	ad2ant2r 746	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → (𝑧 − (1 / 𝑦)) ∈ ℝ)
79	54	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → (1 / 𝑦) ∈ ℝ)
80		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → 𝑧 ∈ ℝ)
81	80	adantr 482	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → 𝑧 ∈ ℝ)
82		simprr 772	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → (1 / 𝑦) < (𝑧 − 𝑥))
83	79, 81, 76, 82	ltsub13d 11817	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → 𝑥 < (𝑧 − (1 / 𝑦)))
84	76, 78, 83	ltled 11359	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → 𝑥 ≤ (𝑧 − (1 / 𝑦)))
85	66	ad2ant2r 746	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → (𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) ↔ (𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ (𝑧 − (1 / 𝑦)))))
86	76, 77, 84, 85	mpbir3and 1343	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) ∧ (𝑦 ∈ ℕ ∧ (1 / 𝑦) < (𝑧 − 𝑥))) → 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))))
87	80, 75	resubcld 11639	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → (𝑧 − 𝑥) ∈ ℝ)
88		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → 𝑥 < 𝑧)
89	75, 80	posdifd 11798	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → (𝑥 < 𝑧 ↔ 0 < (𝑧 − 𝑥)))
90	88, 89	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → 0 < (𝑧 − 𝑥))
91		nnrecl 12467	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 − 𝑥) ∈ ℝ ∧ 0 < (𝑧 − 𝑥)) → ∃𝑦 ∈ ℕ (1 / 𝑦) < (𝑧 − 𝑥))
92	87, 90, 91	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → ∃𝑦 ∈ ℕ (1 / 𝑦) < (𝑧 − 𝑥))
93	86, 92	reximddv 3172	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)) → ∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))))
94	93	ex 414	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ((𝑥 ∈ ℝ ∧ 𝑥 < 𝑧) → ∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦)))))
95	74, 94	impbid 211	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) ↔ (𝑥 ∈ ℝ ∧ 𝑥 < 𝑧)))
96	95, 70	bitr4d 282	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (∃𝑦 ∈ ℕ 𝑥 ∈ (-∞(,](𝑧 − (1 / 𝑦))) ↔ 𝑥 ∈ (-∞(,)𝑧)))
97	45, 96	bitrid 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (𝑥 ∈ ∪ 𝑦 ∈ ℕ (-∞(,](𝑧 − (1 / 𝑦))) ↔ 𝑥 ∈ (-∞(,)𝑧)))
98	97	eqrdv 2731	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∪ 𝑦 ∈ ℕ (-∞(,](𝑧 − (1 / 𝑦))) = (-∞(,)𝑧))
99	98	imaeq2d 6058	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (^◡𝐹 “ ∪ 𝑦 ∈ ℕ (-∞(,](𝑧 − (1 / 𝑦)))) = (^◡𝐹 “ (-∞(,)𝑧)))
100	44, 99	eqtr3id 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))) = (^◡𝐹 “ (-∞(,)𝑧)))
101	1	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → 𝐹:𝐴⟶ℝ)
102		ffun 6718	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℝ → Fun 𝐹)
103		funcnvcnv 6613	. . . . . . . . . . 11 ⊢ (Fun 𝐹 → Fun ^◡^◡𝐹)
104		imadif 6630	. . . . . . . . . . 11 ⊢ (Fun ^◡^◡𝐹 → (^◡𝐹 “ (ℝ ∖ ((𝑧 − (1 / 𝑦))(,)+∞))) = ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞))))
105	101, 102, 103, 104	4syl 19	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (^◡𝐹 “ (ℝ ∖ ((𝑧 − (1 / 𝑦))(,)+∞))) = ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞))))
106	64	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → -∞ ∈ ℝ^*)
107	56	rexrd 11261	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (𝑧 − (1 / 𝑦)) ∈ ℝ^*)
108		pnfxr 11265	. . . . . . . . . . . . . . 15 ⊢ +∞ ∈ ℝ^*
109	108	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → +∞ ∈ ℝ^*)
110	56	mnfltd 13101	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → -∞ < (𝑧 − (1 / 𝑦)))
111	56	ltpnfd 13098	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (𝑧 − (1 / 𝑦)) < +∞)
112		df-ioc 13326	. . . . . . . . . . . . . . 15 ⊢ (,] = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 < 𝑤 ∧ 𝑤 ≤ 𝑣)})
113		df-ioo 13325	. . . . . . . . . . . . . . 15 ⊢ (,) = (𝑢 ∈ ℝ^, 𝑣 ∈ ℝ^ ↦ {𝑤 ∈ ℝ^* ∣ (𝑢 < 𝑤 ∧ 𝑤 < 𝑣)})
114		xrltnle 11278	. . . . . . . . . . . . . . 15 ⊢ (((𝑧 − (1 / 𝑦)) ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((𝑧 − (1 / 𝑦)) < 𝑥 ↔ ¬ 𝑥 ≤ (𝑧 − (1 / 𝑦))))
115		xrlelttr 13132	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ^* ∧ (𝑧 − (1 / 𝑦)) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝑥 ≤ (𝑧 − (1 / 𝑦)) ∧ (𝑧 − (1 / 𝑦)) < +∞) → 𝑥 < +∞))
116		xrlttr 13116	. . . . . . . . . . . . . . 15 ⊢ ((-∞ ∈ ℝ^* ∧ (𝑧 − (1 / 𝑦)) ∈ ℝ^* ∧ 𝑥 ∈ ℝ^*) → ((-∞ < (𝑧 − (1 / 𝑦)) ∧ (𝑧 − (1 / 𝑦)) < 𝑥) → -∞ < 𝑥))
117	112, 113, 114, 113, 115, 116	ixxun 13337	. . . . . . . . . . . . . 14 ⊢ (((-∞ ∈ ℝ^* ∧ (𝑧 − (1 / 𝑦)) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) ∧ (-∞ < (𝑧 − (1 / 𝑦)) ∧ (𝑧 − (1 / 𝑦)) < +∞)) → ((-∞(,](𝑧 − (1 / 𝑦))) ∪ ((𝑧 − (1 / 𝑦))(,)+∞)) = (-∞(,)+∞))
118	106, 107, 109, 110, 111, 117	syl32anc 1379	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((-∞(,](𝑧 − (1 / 𝑦))) ∪ ((𝑧 − (1 / 𝑦))(,)+∞)) = (-∞(,)+∞))
119		uncom 4153	. . . . . . . . . . . . 13 ⊢ ((-∞(,](𝑧 − (1 / 𝑦))) ∪ ((𝑧 − (1 / 𝑦))(,)+∞)) = (((𝑧 − (1 / 𝑦))(,)+∞) ∪ (-∞(,](𝑧 − (1 / 𝑦))))
120		ioomax 13396	. . . . . . . . . . . . 13 ⊢ (-∞(,)+∞) = ℝ
121	118, 119, 120	3eqtr3g 2796	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (((𝑧 − (1 / 𝑦))(,)+∞) ∪ (-∞(,](𝑧 − (1 / 𝑦)))) = ℝ)
122		ioossre 13382	. . . . . . . . . . . . 13 ⊢ ((𝑧 − (1 / 𝑦))(,)+∞) ⊆ ℝ
123		incom 4201	. . . . . . . . . . . . . 14 ⊢ (((𝑧 − (1 / 𝑦))(,)+∞) ∩ (-∞(,](𝑧 − (1 / 𝑦)))) = ((-∞(,](𝑧 − (1 / 𝑦))) ∩ ((𝑧 − (1 / 𝑦))(,)+∞))
124	112, 113, 114	ixxdisj 13336	. . . . . . . . . . . . . . . 16 ⊢ ((-∞ ∈ ℝ^* ∧ (𝑧 − (1 / 𝑦)) ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((-∞(,](𝑧 − (1 / 𝑦))) ∩ ((𝑧 − (1 / 𝑦))(,)+∞)) = ∅)
125	64, 108, 124	mp3an13 1453	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 − (1 / 𝑦)) ∈ ℝ^* → ((-∞(,](𝑧 − (1 / 𝑦))) ∩ ((𝑧 − (1 / 𝑦))(,)+∞)) = ∅)
126	107, 125	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((-∞(,](𝑧 − (1 / 𝑦))) ∩ ((𝑧 − (1 / 𝑦))(,)+∞)) = ∅)
127	123, 126	eqtrid 2785	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (((𝑧 − (1 / 𝑦))(,)+∞) ∩ (-∞(,](𝑧 − (1 / 𝑦)))) = ∅)
128		uneqdifeq 4492	. . . . . . . . . . . . 13 ⊢ ((((𝑧 − (1 / 𝑦))(,)+∞) ⊆ ℝ ∧ (((𝑧 − (1 / 𝑦))(,)+∞) ∩ (-∞(,](𝑧 − (1 / 𝑦)))) = ∅) → ((((𝑧 − (1 / 𝑦))(,)+∞) ∪ (-∞(,](𝑧 − (1 / 𝑦)))) = ℝ ↔ (ℝ ∖ ((𝑧 − (1 / 𝑦))(,)+∞)) = (-∞(,](𝑧 − (1 / 𝑦)))))
129	122, 127, 128	sylancr 588	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((((𝑧 − (1 / 𝑦))(,)+∞) ∪ (-∞(,](𝑧 − (1 / 𝑦)))) = ℝ ↔ (ℝ ∖ ((𝑧 − (1 / 𝑦))(,)+∞)) = (-∞(,](𝑧 − (1 / 𝑦)))))
130	121, 129	mpbid 231	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (ℝ ∖ ((𝑧 − (1 / 𝑦))(,)+∞)) = (-∞(,](𝑧 − (1 / 𝑦))))
131	130	imaeq2d 6058	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (^◡𝐹 “ (ℝ ∖ ((𝑧 − (1 / 𝑦))(,)+∞))) = (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))))
132	105, 131	eqtr3d 2775	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞))) = (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))))
133	42	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (^◡𝐹 “ ℝ) ∈ dom vol)
134		oveq1 7413	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑧 − (1 / 𝑦)) → (𝑥(,)+∞) = ((𝑧 − (1 / 𝑦))(,)+∞))
135	134	imaeq2d 6058	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑧 − (1 / 𝑦)) → (^◡𝐹 “ (𝑥(,)+∞)) = (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞)))
136	135	eleq1d 2819	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 − (1 / 𝑦)) → ((^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol ↔ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞)) ∈ dom vol))
137	32	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ∀𝑥 ∈ ℝ (^◡𝐹 “ (𝑥(,)+∞)) ∈ dom vol)
138	136, 137, 56	rspcdva 3614	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞)) ∈ dom vol)
139		difmbl 25052	. . . . . . . . . 10 ⊢ (((^◡𝐹 “ ℝ) ∈ dom vol ∧ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞)) ∈ dom vol) → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞))) ∈ dom vol)
140	133, 138, 139	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → ((^◡𝐹 “ ℝ) ∖ (^◡𝐹 “ ((𝑧 − (1 / 𝑦))(,)+∞))) ∈ dom vol)
141	132, 140	eqeltrrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ ℝ) ∧ 𝑦 ∈ ℕ) → (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))) ∈ dom vol)
142	141	ralrimiva 3147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∀𝑦 ∈ ℕ (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))) ∈ dom vol)
143		iunmbl 25062	. . . . . . 7 ⊢ (∀𝑦 ∈ ℕ (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))) ∈ dom vol → ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))) ∈ dom vol)
144	142, 143	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → ∪ 𝑦 ∈ ℕ (^◡𝐹 “ (-∞(,](𝑧 − (1 / 𝑦)))) ∈ dom vol)
145	100, 144	eqeltrrd 2835	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ ℝ) → (^◡𝐹 “ (-∞(,)𝑧)) ∈ dom vol)
146	145	ralrimiva 3147	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ ℝ (^◡𝐹 “ (-∞(,)𝑧)) ∈ dom vol)
147		oveq2 7414	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (-∞(,)𝑧) = (-∞(,)𝑥))
148	147	imaeq2d 6058	. . . . . 6 ⊢ (𝑧 = 𝑥 → (^◡𝐹 “ (-∞(,)𝑧)) = (^◡𝐹 “ (-∞(,)𝑥)))
149	148	eleq1d 2819	. . . . 5 ⊢ (𝑧 = 𝑥 → ((^◡𝐹 “ (-∞(,)𝑧)) ∈ dom vol ↔ (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol))
150	149	cbvralvw 3235	. . . 4 ⊢ (∀𝑧 ∈ ℝ (^◡𝐹 “ (-∞(,)𝑧)) ∈ dom vol ↔ ∀𝑥 ∈ ℝ (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
151	146, 150	sylib 217	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
152	151	r19.21bi 3249	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (^◡𝐹 “ (-∞(,)𝑥)) ∈ dom vol)
153	1, 43, 31, 152	ismbf2d 25149	1 ⊢ (𝜑 → 𝐹 ∈ MblFn)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ∪ ciun 4997 class class class wbr 5148 ^◡ccnv 5675 dom cdm 5676 “ cima 5679 Fun wfun 6535 ⟶wf 6537 (class class class)co 7406 ℝcr 11106 0cc0 11107 1c1 11108 +∞cpnf 11242 -∞cmnf 11243 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 − cmin 11441 -cneg 11442 / cdiv 11868 ℕcn 12209 ℝ⁺crp 12971 (,)cioo 13321 (,]cioc 13322 volcvol 24972 MblFncmbf 25123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cc 10427 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-dju 9893 df-card 9931 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xadd 13090 df-ioo 13325 df-ioc 13326 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-xmet 20930 df-met 20931 df-ovol 24973 df-vol 24974 df-mbf 25128

This theorem is referenced by: mbfaddlem 25169 mbfsup 25173

W3C validator