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Theorem ibladdlem 25875

Description: Lemma for ibladd 25876. (Contributed by Mario Carneiro, 17-Aug-2014.)

**Hypotheses**
Ref	Expression
ibladd.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
ibladd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
ibladd.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 = (𝐵 + 𝐶))
ibladd.4	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
ibladd.5	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
ibladd.6	⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
ibladd.7	⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)

**Assertion**
Ref	Expression
ibladdlem	⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑥) 𝐷(𝑥)

**Proof of Theorem ibladdlem**
Step	Hyp	Ref	Expression
1		ifan 4601	. . . 4 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0)
2		ibladd.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 = (𝐵 + 𝐶))
3		ibladd.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4		ibladd.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
5	3, 4	readdcld 11319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℝ)
6	2, 5	eqeltrd 2844	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ ℝ)
7		0re 11292	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		ifcl 4593	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
9	6, 7, 8	sylancl 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
10	9	rexrd 11340	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ^*)
11		max1 13247	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
12	7, 6, 11	sylancr 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
13		elxrge0 13517	. . . . . . 7 ⊢ (if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞) ↔ (if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ 𝐷, 𝐷, 0)))
14	10, 12, 13	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞))
15		0e0iccpnf 13519	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
16	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
17	14, 16	ifclda 4583	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
18	17	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
19	1, 18	eqeltrid 2848	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
20	19	fmpttd 7149	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
21		reex 11275	. . . . . . . 8 ⊢ ℝ ∈ V
22	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
23		ifan 4601	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
24		ifcl 4593	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
25	3, 7, 24	sylancl 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
26	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
27	25, 26	ifclda 4583	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
28	23, 27	eqeltrid 2848	. . . . . . . 8 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
30		ifan 4601	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
31		ifcl 4593	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
32	4, 7, 31	sylancl 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
33	32, 26	ifclda 4583	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
34	30, 33	eqeltrid 2848	. . . . . . . 8 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
35	34	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
36		eqidd 2741	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
37		eqidd 2741	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
38	22, 29, 35, 36, 37	offval2 7734	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) = (𝑥 ∈ ℝ ↦ (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))))
39		iftrue 4554	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
40		ibar 528	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ 𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)))
41	40	ifbid 4571	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ 𝐵, 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
42		ibar 528	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ 𝐶 ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
43	42	ifbid 4571	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
44	41, 43	oveq12d 7466	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))
45	39, 44	eqtr2d 2781	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
46		00id 11465	. . . . . . . . 9 ⊢ (0 + 0) = 0
47		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) → 𝑥 ∈ 𝐴)
48	47	con3i 154	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵))
49	48	iffalsed 4559	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
50		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶) → 𝑥 ∈ 𝐴)
51	50	con3i 154	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))
52	51	iffalsed 4559	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
53	49, 52	oveq12d 7466	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
54		iffalse 4557	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
55	46, 53, 54	3eqtr4a 2806	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
56	45, 55	pm2.61i 182	. . . . . . 7 ⊢ (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)
57	56	mpteq2i 5271	. . . . . 6 ⊢ (𝑥 ∈ ℝ ↦ (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
58	38, 57	eqtrdi 2796	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
59	58	fveq2d 6924	. . . 4 ⊢ (𝜑 → (∫₂‘((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) = (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))))
60		ibladd.4	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
61	60, 3	mbfdm2 25691	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom vol)
62		mblss 25585	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
63	61, 62	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
64		rembl 25594	. . . . . . 7 ⊢ ℝ ∈ dom vol
65	64	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ dom vol)
66	28	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
67		eldifn 4155	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
68	67	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
69	68	intnanrd 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵))
70	69	iffalsed 4559	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
71	41	mpteq2ia 5269	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
72	3, 60	mbfpos 25705	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
73	71, 72	eqeltrrid 2849	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
74	63, 65, 66, 70, 73	mbfss 25700	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
75		max1 13247	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
76	7, 3, 75	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
77		elrege0 13514	. . . . . . . . . 10 ⊢ (if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)))
78	25, 76, 77	sylanbrc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞))
79		0e0icopnf 13518	. . . . . . . . . 10 ⊢ 0 ∈ (0[,)+∞)
80	79	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
81	78, 80	ifclda 4583	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
82	23, 81	eqeltrid 2848	. . . . . . 7 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
83	82	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
84	83	fmpttd 7149	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,)+∞))
85		ibladd.6	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
86	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
87	68, 52	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
88	43	mpteq2ia 5269	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) = (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
89		ibladd.5	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn)
90	4, 89	mbfpos 25705	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ MblFn)
91	88, 90	eqeltrrid 2849	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
92	63, 65, 86, 87, 91	mbfss 25700	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
93		max1 13247	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
94	7, 4, 93	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
95		elrege0 13514	. . . . . . . . . 10 ⊢ (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
96	32, 94, 95	sylanbrc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
97	96, 80	ifclda 4583	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
98	30, 97	eqeltrid 2848	. . . . . . 7 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
99	98	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
100	99	fmpttd 7149	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,)+∞))
101		ibladd.7	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
102	74, 84, 85, 92, 100, 101	itg2add 25814	. . . 4 ⊢ (𝜑 → (∫₂‘((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∘_f + (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))))
103	59, 102	eqtr3d 2782	. . 3 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) = ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))))
104	85, 101	readdcld 11319	. . 3 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) ∈ ℝ)
105	103, 104	eqeltrd 2844	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
106	25, 32	readdcld 11319	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
107	106	rexrd 11340	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ^*)
108	25, 32, 76, 94	addge0d 11866	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
109		elxrge0 13517	. . . . . . 7 ⊢ ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞) ↔ ((if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ^* ∧ 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
110	107, 108, 109	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
111	110, 16	ifclda 4583	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
112	111	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
113	112	fmpttd 7149	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
114		max2 13249	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
115	7, 3, 114	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
116		max2 13249	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
117	7, 4, 116	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
118	3, 4, 25, 32, 115, 117	le2addd 11909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
119	2, 118	eqbrtrd 5188	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
120		breq1 5169	. . . . . . . . . . 11 ⊢ (𝐷 = if(0 ≤ 𝐷, 𝐷, 0) → (𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ↔ if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
121		breq1 5169	. . . . . . . . . . 11 ⊢ (0 = if(0 ≤ 𝐷, 𝐷, 0) → (0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ↔ if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))
122	120, 121	ifboth 4587	. . . . . . . . . 10 ⊢ ((𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∧ 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
123	119, 108, 122	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
124		iftrue 4554	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
125	124	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
126	39	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
127	123, 125, 126	3brtr4d 5198	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
128	127	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
129		0le0 12394	. . . . . . . . 9 ⊢ 0 ≤ 0
130	129	a1i 11	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
131		iffalse 4557	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
132	130, 131, 54	3brtr4d 5198	. . . . . . 7 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
133	128, 132	pm2.61d1 180	. . . . . 6 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
134	1, 133	eqbrtrid 5201	. . . . 5 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
135	134	ralrimivw 3156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
136		eqidd 2741	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
137		eqidd 2741	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
138	22, 19, 112, 136, 137	ofrfval2 7735	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)) ↔ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
139	135, 138	mpbird 257	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))
140		itg2le 25794	. . 3 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞) ∧ (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) ∘_r ≤ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))))
141	20, 113, 139, 140	syl3anc 1371	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))))
142		itg2lecl 25793	. 2 ⊢ (((𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ≤ (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)))) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)
143	20, 105, 141, 142	syl3anc 1371	1 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0))) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ∖ cdif 3973 ⊆ wss 3976 ifcif 4548 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ∘_f cof 7712 ∘_r cofr 7713 ℝcr 11183 0cc0 11184 + caddc 11187 +∞cpnf 11321 ℝ^*cxr 11323 ≤ cle 11325 [,)cico 13409 [,]cicc 13410 volcvol 25517 MblFncmbf 25668 ∫₂citg2 25670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-rlim 15535 df-sum 15735 df-rest 17482 df-topgen 17503 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-top 22921 df-topon 22938 df-bases 22974 df-cmp 23416 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-0p 25724

This theorem is referenced by: ibladd 25876

W3C validator