ibladdnclem - Mathbox for Brendan Leahy

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Theorem ibladdnclem 37011

Description: Lemma for ibladdnc 37012; cf ibladdlem 25670, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 37009. (Contributed by Brendan Leahy, 31-Oct-2017.)

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

**Assertion**
Ref	Expression
ibladdnclem	⊢ (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 ibladdnclem**
Step	Hyp	Ref	Expression
1		ifan 4581	. . . 4 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0)
2		ibladdnclem.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 = (𝐵 + 𝐶))
3		ibladdnclem.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
4		ibladdnclem.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ)
5	3, 4	readdcld 11250	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ∈ ℝ)
6	2, 5	eqeltrd 2832	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ∈ ℝ)
7		0re 11223	. . . . . . . . 9 ⊢ 0 ∈ ℝ
8		ifcl 4573	. . . . . . . . 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 11271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ∈ ℝ^*)
11		max1 13171	. . . . . . . 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 13441	. . . . . . 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 13443	. . . . . . 7 ⊢ 0 ∈ (0[,]+∞)
16	15	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,]+∞))
17	14, 16	ifclda 4563	. . . . 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 2836	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ∈ (0[,]+∞))
20	19	fmpttd 7116	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)):ℝ⟶(0[,]+∞))
21		reex 11207	. . . . . . . 8 ⊢ ℝ ∈ V
22	21	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ V)
23		ifan 4581	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0)
24		ifcl 4573	. . . . . . . . . . 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 4563	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ ℝ)
28	23, 27	eqeltrid 2836	. . . . . . . 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 4581	. . . . . . . . 9 ⊢ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0)
31		ifcl 4573	. . . . . . . . . . 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 4563	. . . . . . . . 9 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ ℝ)
34	30, 33	eqeltrid 2836	. . . . . . . 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 2732	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)))
37		eqidd 2732	. . . . . . 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 7694	. . . . . 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 4534	. . . . . . . . 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 4551	. . . . . . . . . 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 4551	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(0 ≤ 𝐶, 𝐶, 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))
44	41, 43	oveq12d 7430	. . . . . . . . 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 2772	. . . . . . . 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 11396	. . . . . . . . 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 4539	. . . . . . . . . 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 4539	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ 𝐴 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) = 0)
53	49, 52	oveq12d 7430	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → (if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) + if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)) = (0 + 0))
54		iffalse 4537	. . . . . . . . 9 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) = 0)
55	46, 53, 54	3eqtr4a 2797	. . . . . . . 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 5253	. . . . . 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 2787	. . . . 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 6895	. . . 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		ibladdnclem.4	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
61	60, 3	mbfdm2 25487	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom vol)
62		mblss 25381	. . . . . . 7 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ)
63	61, 62	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
64		rembl 25390	. . . . . . 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 4127	. . . . . . . . 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 4539	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℝ ∖ 𝐴)) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0) = 0)
71	41	mpteq2ia 5251	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))
72	3, 60	mbfpos 25501	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn)
73	71, 72	eqeltrrid 2837	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
74	63, 65, 66, 70, 73	mbfss 25496	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)) ∈ MblFn)
75		max1 13171	. . . . . . . . . . 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 13438	. . . . . . . . . 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 13442	. . . . . . . . . 10 ⊢ 0 ∈ (0[,)+∞)
80	79	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑥 ∈ 𝐴) → 0 ∈ (0[,)+∞))
81	78, 80	ifclda 4563	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐵, 𝐵, 0), 0) ∈ (0[,)+∞))
82	23, 81	eqeltrid 2836	. . . . . . 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 7116	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0)):ℝ⟶(0[,)+∞))
85		ibladdnclem.6	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) ∈ ℝ)
86		max1 13171	. . . . . . . . . . 11 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
87	7, 4, 86	sylancr 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ if(0 ≤ 𝐶, 𝐶, 0))
88		elrege0 13438	. . . . . . . . . 10 ⊢ (if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞) ↔ (if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ ∧ 0 ≤ if(0 ≤ 𝐶, 𝐶, 0)))
89	32, 87, 88	sylanbrc 582	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ (0[,)+∞))
90	89, 80	ifclda 4563	. . . . . . . 8 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐶, 𝐶, 0), 0) ∈ (0[,)+∞))
91	30, 90	eqeltrid 2836	. . . . . . 7 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
92	91	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0) ∈ (0[,)+∞))
93	92	fmpttd 7116	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)):ℝ⟶(0[,)+∞))
94		ibladdnclem.7	. . . . 5 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0))) ∈ ℝ)
95	74, 84, 85, 93, 94	itg2addnc 37009	. . . 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)))))
96	59, 95	eqtr3d 2773	. . 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)))))
97	85, 94	readdcld 11250	. . 3 ⊢ (𝜑 → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵), 𝐵, 0))) + (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶), 𝐶, 0)))) ∈ ℝ)
98	96, 97	eqeltrd 2832	. 2 ⊢ (𝜑 → (∫₂‘(𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))) ∈ ℝ)
99	25, 32	readdcld 11250	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ)
100	99	rexrd 11271	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ^*)
101	25, 32, 76, 87	addge0d 11797	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 0 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
102		elxrge0 13441	. . . . . . 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))))
103	100, 101, 102	sylanbrc 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ (0[,]+∞))
104	103, 16	ifclda 4563	. . . . 5 ⊢ (𝜑 → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
105	104	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0) ∈ (0[,]+∞))
106	105	fmpttd 7116	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0)):ℝ⟶(0[,]+∞))
107		max2 13173	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
108	7, 3, 107	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ if(0 ≤ 𝐵, 𝐵, 0))
109		max2 13173	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ 𝐶 ∈ ℝ) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
110	7, 4, 109	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ≤ if(0 ≤ 𝐶, 𝐶, 0))
111	3, 4, 25, 32, 108, 110	le2addd 11840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 + 𝐶) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
112	2, 111	eqbrtrd 5170	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐷 ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
113		breq1 5151	. . . . . . . . . . 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))))
114		breq1 5151	. . . . . . . . . . 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))))
115	113, 114	ifboth 4567	. . . . . . . . . 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)))
116	112, 101, 115	syl2anc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐷, 𝐷, 0) ≤ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))
117		iftrue 4534	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
118	117	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = if(0 ≤ 𝐷, 𝐷, 0))
119	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)))
120	116, 118, 119	3brtr4d 5180	. . . . . . . 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))
121	120	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)))
122		0le0 12320	. . . . . . . . 9 ⊢ 0 ≤ 0
123	122	a1i 11	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → 0 ≤ 0)
124		iffalse 4537	. . . . . . . 8 ⊢ (¬ 𝑥 ∈ 𝐴 → if(𝑥 ∈ 𝐴, if(0 ≤ 𝐷, 𝐷, 0), 0) = 0)
125	123, 124, 54	3brtr4d 5180	. . . . . . 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))
126	121, 125	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))
127	1, 126	eqbrtrid 5183	. . . . 5 ⊢ (𝜑 → if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0) ≤ if(𝑥 ∈ 𝐴, (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)), 0))
128	127	ralrimivw 3149	. . . 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))
129		eqidd 2732	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐷), 𝐷, 0)))
130		eqidd 2732	. . . . 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)))
131	22, 19, 105, 129, 130	ofrfval2 7695	. . . 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)))
132	128, 131	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)))
133		itg2le 25590	. . 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))))
134	20, 106, 132, 133	syl3anc 1370	. 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))))
135		itg2lecl 25589	. 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)
136	20, 98, 134, 135	syl3anc 1370	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 1540 ∈ wcel 2105 ∀wral 3060 Vcvv 3473 ∖ cdif 3945 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ∘_f cof 7672 ∘_r cofr 7673 ℝcr 11115 0cc0 11116 + caddc 11119 +∞cpnf 11252 ℝ^*cxr 11254 ≤ cle 11256 [,)cico 13333 [,]cicc 13334 volcvol 25313 MblFncmbf 25464 ∫₂citg2 25466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-ofr 7675 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-2o 8473 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ioo 13335 df-ico 13337 df-icc 13338 df-fz 13492 df-fzo 13635 df-fl 13764 df-seq 13974 df-exp 14035 df-hash 14298 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-sum 15640 df-rest 17375 df-topgen 17396 df-psmet 21226 df-xmet 21227 df-met 21228 df-bl 21229 df-mopn 21230 df-top 22717 df-topon 22734 df-bases 22770 df-cmp 23212 df-ovol 25314 df-vol 25315 df-mbf 25469 df-itg1 25470 df-itg2 25471

This theorem is referenced by: ibladdnc 37012

W3C validator