ibladdlem - Metamath Proof Explorer

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

Theorem ibladdlem 24717

Description: Lemma for ibladd 24718. (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 4492	. . . 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 10862	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℝ)
6	2, 5	eqeltrd 2838	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ ℝ)
7		0re 10835	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		ifcl 4484	. . . . . . . . 9 ⊢ ((𝐷 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
9	6, 7, 8	sylancl 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ)
10	9	rexrd 10883	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ^*)
11		max1 12775	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 𝐷 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
12	7, 6, 11	sylancr 590	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐷, 𝐷, 0))
13		elxrge0 13045	. . . . . . 7 ⊢ (if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞) ↔ (if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ^* ∧ 0 ≤ if(0 ≤ 𝐷, 𝐷, 0)))
14	10, 12, 13	sylanbrc 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ (0[,]+∞))
15		0e0iccpnf 13047	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
16	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
17	14, 16	ifclda 4474	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
18	17	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) ∈ (0[,]+∞))
19	1, 18	eqeltrid 2842	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
20	19	fmpttd 6932	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
21		reex 10820	. . . . . . . 8 ⊢ ℝ ∈ V
22	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
23		ifan 4492	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
24		ifcl 4484	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
25	3, 7, 24	sylancl 589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ)
26	7	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ ℝ)
27	25, 26	ifclda 4474	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
28	23, 27	eqeltrid 2842	. . . . . . . 8 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
29	28	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
30		ifan 4492	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
31		ifcl 4484	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ ℝ ∧ 0 ∈ ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
32	4, 7, 31	sylancl 589	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ)
33	32, 26	ifclda 4474	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
34	30, 33	eqeltrid 2842	. . . . . . . 8 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
35	34	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ ℝ)
36		eqidd 2738	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
37		eqidd 2738	. . . . . . 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 7488	. . . . . 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 4445	. . . . . . . . 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 532	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ 𝐵 ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)))
41	40	ifbid 4462	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ 𝐵, 𝐵, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
42		ibar 532	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (0 ≤ 𝐶 ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)))
43	42	ifbid 4462	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
44	41, 43	oveq12d 7231	. . . . . . . . 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 2778	. . . . . . . 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 11007	. . . . . . . . 9 ⊢ (0 + 0) = 0
47		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) → 𝑥 ∈ 𝐴)
48	47	con3i 157	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵))
49	48	iffalsed 4450	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
50		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶) → 𝑥 ∈ 𝐴)
51	50	con3i 157	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ 𝐴 → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))
52	51	iffalsed 4450	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
53	49, 52	oveq12d 7231	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
54		iffalse 4448	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
55	46, 53, 54	3eqtr4a 2804	. . . . . . . 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 185	. . . . . . 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 5147	. . . . . 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 2794	. . . . 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 6721	. . . 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 24534	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom vol)
62		mblss 24428	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
63	61, 62	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
64		rembl 24437	. . . . . . 7 ⊢ ℝ ∈ dom vol
65	64	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ dom vol)
66	28	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ ℝ)
67		eldifn 4042	. . . . . . . . 9 ⊢ (𝑥 ∈ (ℝ ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
68	67	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
69	68	intnanrd 493	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → ¬ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵))
70	69	iffalsed 4450	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
71	41	mpteq2ia 5146	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
72	3, 60	mbfpos 24548	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
73	71, 72	eqeltrrid 2843	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
74	63, 65, 66, 70, 73	mbfss 24543	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
75		max1 12775	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
76	7, 3, 75	sylancr 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐵, 𝐵, 0))
77		elrege0 13042	. . . . . . . . . 10 ⊢ (if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐵, 𝐵, 0)))
78	25, 76, 77	sylanbrc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ (0[,)+∞))
79		0e0icopnf 13046	. . . . . . . . . 10 ⊢ 0 ∈ (0[,)+∞)
80	79	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
81	78, 80	ifclda 4474	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
82	23, 81	eqeltrid 2842	. . . . . . 7 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
83	82	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) ∈ (0[,)+∞))
84	83	fmpttd 6932	. . . . 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 484	. . . . . 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 5146	. . . . . . 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 24548	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐶, 𝐶, 0)) ∈ MblFn)
91	88, 90	eqeltrrid 2843	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
92	63, 65, 86, 87, 91	mbfss 24543	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) ∈ MblFn)
93		max1 12775	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
94	7, 4, 93	sylancr 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
95		elrege0 13042	. . . . . . . . . 10 ⊢ (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
96	32, 94, 95	sylanbrc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
97	96, 80	ifclda 4474	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
98	30, 97	eqeltrid 2842	. . . . . . 7 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
99	98	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
100	99	fmpttd 6932	. . . . 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 24657	. . . 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 2779	. . 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 10862	. . 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 2838	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
106	25, 32	readdcld 10862	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
107	106	rexrd 10883	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ^*)
108	25, 32, 76, 94	addge0d 11408	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
109		elxrge0 13045	. . . . . . 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 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
111	110, 16	ifclda 4474	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
112	111	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
113	112	fmpttd 6932	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
114		max2 12777	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
115	7, 3, 114	sylancr 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
116		max2 12777	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
117	7, 4, 116	sylancr 590	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
118	3, 4, 25, 32, 115, 117	le2addd 11451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
119	2, 118	eqbrtrd 5075	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
120		breq1 5056	. . . . . . . . . . 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 5056	. . . . . . . . . . 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 4478	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
124		iftrue 4445	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
125	124	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
126	39	adantl 485	. . . . . . . . 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 5085	. . . . . . . 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 416	. . . . . . 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 11931	. . . . . . . . 9 ⊢ 0 ≤ 0
130	129	a1i 11	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
131		iffalse 4448	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
132	130, 131, 54	3brtr4d 5085	. . . . . . 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 183	. . . . . 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 5088	. . . . 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 3106	. . . 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 2738	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
137		eqidd 2738	. . . . 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 7489	. . . 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 260	. . 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 24637	. . 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 1373	. 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 24636	. 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 1373	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 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 Vcvv 3408 ∖ cdif 3863 ⊆ wss 3866 ifcif 4439 class class class wbr 5053 ↦ cmpt 5135 dom cdm 5551 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ∘_f cof 7467 ∘_r cofr 7468 ℝcr 10728 0cc0 10729 + caddc 10732 +∞cpnf 10864 ℝ^*cxr 10866 ≤ cle 10868 [,)cico 12937 [,]cicc 12938 volcvol 24360 MblFncmbf 24511 ∫₂citg2 24513

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cc 10049 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-disj 5019 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-ofr 7470 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-omul 8207 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-acn 9558 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ioo 12939 df-ioc 12940 df-ico 12941 df-icc 12942 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-top 21791 df-topon 21808 df-bases 21843 df-cmp 22284 df-ovol 24361 df-vol 24362 df-mbf 24516 df-itg1 24517 df-itg2 24518 df-0p 24567

This theorem is referenced by: ibladd 24718

W3C validator