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Theorem bddmulibl 25901

Description: A bounded function times an integrable function is integrable. (Contributed by Mario Carneiro, 12-Aug-2014.)

**Assertion**
Ref	Expression
bddmulibl	⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹)

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐺,𝑦

Proof of Theorem bddmulibl Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbff 25687	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
2	1	ad2antrr 736	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹:dom 𝐹⟶ℂ)
3	2	ffnd 6692	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 Fn dom 𝐹)
4		iblmbf 25829	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
5	4	ad2antlr 737	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ MblFn)
6		mbff 25687	. . . . . . 7 ⊢ (𝐺 ∈ MblFn → 𝐺:dom 𝐺⟶ℂ)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺:dom 𝐺⟶ℂ)
8	7	ffnd 6692	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 Fn dom 𝐺)
9		mbfdm 25688	. . . . . 6 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
10	9	ad2antrr 736	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐹 ∈ dom vol)
11		mbfdm 25688	. . . . . 6 ⊢ (𝐺 ∈ MblFn → dom 𝐺 ∈ dom vol)
12	5, 11	syl 17	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐺 ∈ dom vol)
13		eqid 2762	. . . . 5 ⊢ (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14		eqidd 2763	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
15		eqidd 2763	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) = (𝐺‘𝑧))
16	3, 8, 10, 12, 13, 14, 15	offval 7669	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) = (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))))
17		ovexd 7431	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ∈ V)
18		simpll 776	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 ∈ MblFn)
19	18, 5	mbfmul 25788	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ MblFn)
20	16, 19	eqeltrrd 2863	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ MblFn)
21		absf 15365	. . . . . . . . 9 ⊢ abs:ℂ⟶ℝ
22	21	a1i 11	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → abs:ℂ⟶ℝ)
23	20, 17	mbfmptcl 25698	. . . . . . . 8 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ∈ ℂ)
24	22, 23	cofmpt 7114	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (abs ∘ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) = (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))))
25	23	fmpttd 7096	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))):(dom 𝐹 ∩ dom 𝐺)⟶ℂ)
26		ax-resscn 11130	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
27		ssid 3958	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
28		cncfss 24961	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
29	26, 27, 28	mp2an 702	. . . . . . . . . 10 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
30		abscncf 24963	. . . . . . . . . 10 ⊢ abs ∈ (ℂ–cn→ℝ)
31	29, 30	sselii 3933	. . . . . . . . 9 ⊢ abs ∈ (ℂ–cn→ℂ)
32	31	a1i 11	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → abs ∈ (ℂ–cn→ℂ))
33		cncombf 25720	. . . . . . . 8 ⊢ (((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ MblFn ∧ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))):(dom 𝐹 ∩ dom 𝐺)⟶ℂ ∧ abs ∈ (ℂ–cn→ℂ)) → (abs ∘ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn)
34	20, 25, 32, 33	syl3anc 1390	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (abs ∘ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn)
35	24, 34	eqeltrrd 2863	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn)
36	23	abscld 15466	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ)
37	36	rexrd 11232	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ^*)
38	23	absge0d 15474	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))))
39		elxrge0 13461	. . . . . . . . . . 11 ⊢ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))))
40	37, 38, 39	sylanbrc 592	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞))
41		0e0iccpnf 13463	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,]+∞)
42	41	a1i 11	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ∈ (0[,]+∞))
43	40, 42	ifclda 4516	. . . . . . . . 9 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ∈ (0[,]+∞))
44	43	adantr 484	. . . . . . . 8 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ∈ (0[,]+∞))
45	44	fmpttd 7096	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞))
46		reex 11164	. . . . . . . . . . . . . . 15 ⊢ ℝ ∈ V
47	46	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → ℝ ∈ V)
48		simprl 780	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℝ)
49	48	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
50		elinel2 4154	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐺)
51		ffvelcdm 7062	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
52	7, 50, 51	syl2an 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐺‘𝑧) ∈ ℂ)
53	52	abscld 15466	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐺‘𝑧)) ∈ ℝ)
54	52	absge0d 15474	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (abs‘(𝐺‘𝑧)))
55		elrege0 13458	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘(𝐺‘𝑧)) ∈ (0[,)+∞) ↔ ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
56	53, 54, 55	sylanbrc 592	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐺‘𝑧)) ∈ (0[,)+∞))
57		0e0icopnf 13462	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ (0[,)+∞)
58	57	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ∈ (0[,)+∞))
59	56, 58	ifclda 4516	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0) ∈ (0[,)+∞))
60	59	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0) ∈ (0[,)+∞))
61		fconstmpt 5709	. . . . . . . . . . . . . . 15 ⊢ (ℝ × {𝑥}) = (𝑧 ∈ ℝ ↦ 𝑥)
62	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (ℝ × {𝑥}) = (𝑧 ∈ ℝ ↦ 𝑥))
63		eqidd 2763	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))
64	47, 49, 60, 62, 63	offval2 7680	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((ℝ × {𝑥}) ∘_f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) = (𝑧 ∈ ℝ ↦ (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))))
65		ovif2 7495	. . . . . . . . . . . . . . 15 ⊢ (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)) = if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), (𝑥 · 0))
66	48	recnd 11210	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℂ)
67	66	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℂ)
68	67	mul01d 11382	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · 0) = 0)
69	68	ifeq2d 4501	. . . . . . . . . . . . . . 15 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), (𝑥 · 0)) = if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
70	65, 69	eqtrid 2809	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)) = if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
71	70	mpteq2dv 5194	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑧 ∈ ℝ ↦ (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))
72	64, 71	eqtrd 2797	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((ℝ × {𝑥}) ∘_f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))
73	72	fveq2d 6871	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∫₂‘((ℝ × {𝑥}) ∘_f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) = (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))))
74	59	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0) ∈ (0[,)+∞))
75	74	fmpttd 7096	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)):ℝ⟶(0[,)+∞))
76	75	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)):ℝ⟶(0[,)+∞))
77		inss2 4189	. . . . . . . . . . . . . . . . . 18 ⊢ (dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐺
78	77	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐺)
79	20, 17	mbfdm2 25699	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
80	7	ffvelcdmda 7065	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
81	7	feqmptd 6935	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 = (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)))
82		simplr 778	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ 𝐿¹)
83	81, 82	eqeltrrd 2863	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)) ∈ 𝐿¹)
84	78, 79, 80, 83	iblss 25867	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺‘𝑧)) ∈ 𝐿¹)
85	52, 84	iblabs 25891	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ 𝐿¹)
86	53, 54	iblpos 25855	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ 𝐿¹ ↔ ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ MblFn ∧ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ)))
87	85, 86	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ MblFn ∧ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ))
88	87	simprd 499	. . . . . . . . . . . . 13 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ)
89	88	adantr 484	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ)
90		simplrl 786	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℝ)
91		neq0 4304	. . . . . . . . . . . . . . 15 ⊢ (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
92		0re 11183	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ
93	92	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ∈ ℝ)
94		elinel1 4153	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐹)
95		ffvelcdm 7062	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
96	2, 94, 95	syl2an 605	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐹‘𝑧) ∈ ℂ)
97	96	abscld 15466	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐹‘𝑧)) ∈ ℝ)
98		simplrl 786	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 𝑥 ∈ ℝ)
99	96	absge0d 15474	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (abs‘(𝐹‘𝑧)))
100		simprr 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)
101		2fveq3 6872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑧)))
102	101	breq1d 5110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → ((abs‘(𝐹‘𝑦)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑧)) ≤ 𝑥))
103	102	rspccva 3580	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 ∧ 𝑧 ∈ dom 𝐹) → (abs‘(𝐹‘𝑧)) ≤ 𝑥)
104	100, 94, 103	syl2an 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐹‘𝑧)) ≤ 𝑥)
105	93, 97, 98, 99, 104	letrd 11340	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ 𝑥)
106	105	ex 416	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥))
107	106	exlimdv 1953	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥))
108	91, 107	biimtrid 244	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ → 0 ≤ 𝑥))
109	108	imp 410	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 0 ≤ 𝑥)
110		elrege0 13458	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
111	90, 109, 110	sylanbrc 592	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ (0[,)+∞))
112	76, 89, 111	itg2mulc 25809	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∫₂‘((ℝ × {𝑥}) ∘_f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) = (𝑥 · (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))))
113	73, 112	eqtr3d 2799	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (𝑥 · (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))))
114	90, 89	remulcld 11212	. . . . . . . . . 10 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) ∈ ℝ)
115	113, 114	eqeltrd 2862	. . . . . . . . 9 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ)
116	115	ex 416	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ))
117		noel 4290	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑧 ∈ ∅
118		eleq2 2851	. . . . . . . . . . . . . 14 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↔ 𝑧 ∈ ∅))
119	117, 118	mtbiri 329	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
120		iffalse 4489	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = 0)
121	119, 120	syl 17	. . . . . . . . . . . 12 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = 0)
122	121	mpteq2dv 5194	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ 0))
123		fconstmpt 5709	. . . . . . . . . . 11 ⊢ (ℝ × {0}) = (𝑧 ∈ ℝ ↦ 0)
124	122, 123	eqtr4di 2815	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (ℝ × {0}))
125	124	fveq2d 6871	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (∫₂‘(ℝ × {0})))
126		itg20 25799	. . . . . . . . . 10 ⊢ (∫₂‘(ℝ × {0})) = 0
127	126, 92	eqeltri 2858	. . . . . . . . 9 ⊢ (∫₂‘(ℝ × {0})) ∈ ℝ
128	125, 127	eqeltrdi 2870	. . . . . . . 8 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ)
129	116, 128	pm2.61d2 182	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ)
130	98, 53	remulcld 11212	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ)
131	130	rexrd 11232	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ^*)
132	98, 53, 105, 54	mulge0d 11764	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (𝑥 · (abs‘(𝐺‘𝑧))))
133		elxrge0 13461	. . . . . . . . . . . 12 ⊢ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (𝑥 · (abs‘(𝐺‘𝑧)))))
134	131, 132, 133	sylanbrc 592	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞))
135	134, 42	ifclda 4516	. . . . . . . . . 10 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) ∈ (0[,]+∞))
136	135	adantr 484	. . . . . . . . 9 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) ∈ (0[,]+∞))
137	136	fmpttd 7096	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞))
138	96, 52	absmuld 15484	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) = ((abs‘(𝐹‘𝑧)) · (abs‘(𝐺‘𝑧))))
139		abscl 15305	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → (abs‘(𝐺‘𝑧)) ∈ ℝ)
140		absge0 15314	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → 0 ≤ (abs‘(𝐺‘𝑧)))
141	139, 140	jca 519	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑧) ∈ ℂ → ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
142	52, 141	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
143		lemul1a 12045	. . . . . . . . . . . . . 14 ⊢ ((((abs‘(𝐹‘𝑧)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧)))) ∧ (abs‘(𝐹‘𝑧)) ≤ 𝑥) → ((abs‘(𝐹‘𝑧)) · (abs‘(𝐺‘𝑧))) ≤ (𝑥 · (abs‘(𝐺‘𝑧))))
144	97, 98, 142, 104, 143	syl31anc 1392	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((abs‘(𝐹‘𝑧)) · (abs‘(𝐺‘𝑧))) ≤ (𝑥 · (abs‘(𝐺‘𝑧))))
145	138, 144	eqbrtrd 5122	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ≤ (𝑥 · (abs‘(𝐺‘𝑧))))
146		iftrue 4486	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))))
147	146	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))))
148		iftrue 4486	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧))))
149	148	adantl 485	. . . . . . . . . . . 12 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧))))
150	145, 147, 149	3brtr4d 5132	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
151		0le0 12319	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
152	151	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 0)
153		iffalse 4489	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = 0)
154	152, 153, 120	3brtr4d 5132	. . . . . . . . . . . 12 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
155	154	adantl 485	. . . . . . . . . . 11 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
156	150, 155	pm2.61dan 822	. . . . . . . . . 10 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
157	156	ralrimivw 3158	. . . . . . . . 9 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑧 ∈ ℝ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))
158	46	a1i 11	. . . . . . . . . 10 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ℝ ∈ V)
159		eqidd 2763	. . . . . . . . . 10 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)))
160		eqidd 2763	. . . . . . . . . 10 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))
161	158, 44, 136, 159, 160	ofrfval2 7681	. . . . . . . . 9 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) ∘_r ≤ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) ↔ ∀𝑧 ∈ ℝ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))
162	157, 161	mpbird 259	. . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) ∘_r ≤ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))
163		itg2le 25801	. . . . . . . 8 ⊢ (((𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞) ∧ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) ∘_r ≤ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ≤ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))))
164	45, 137, 162, 163	syl3anc 1390	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ≤ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))))
165		itg2lecl 25800	. . . . . . 7 ⊢ (((𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞) ∧ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ ∧ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ≤ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ∈ ℝ)
166	45, 129, 164, 165	syl3anc 1390	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ∈ ℝ)
167	36, 38	iblpos 25855	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ 𝐿¹ ↔ ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn ∧ (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ∈ ℝ)))
168	35, 166, 167	mpbir2and 723	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ 𝐿¹)
169	17, 20, 168	iblabsr 25892	. . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ 𝐿¹)
170	16, 169	eqeltrd 2862	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹)
171	170	rexlimdvaa 3164	. 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹))
172	171	3impia 1130	1 ⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∃wex 1799 ∈ wcel 2142 ∀wral 3076 ∃wrex 3086 Vcvv 3454 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 ifcif 4480 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 × cxp 5645 dom cdm 5647 ∘ ccom 5651 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 ∘_f cof 7658 ∘_r cofr 7659 ℂcc 11071 ℝcr 11072 0cc0 11073 · cmul 11078 +∞cpnf 11213 ℝ^*cxr 11215 ≤ cle 11217 [,)cico 13351 [,]cicc 13352 abscabs 15261 –cn→ccncf 24938 volcvol 25525 MblFncmbf 25676 ∫₂citg2 25678 𝐿¹cibl 25679

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cc 10392 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-ofr 7661 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-dju 9859 df-card 9897 df-acn 9900 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-ioc 13354 df-ico 13355 df-icc 13356 df-fz 13513 df-fzo 13660 df-fl 13802 df-mod 13880 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-limsup 15498 df-clim 15515 df-rlim 15516 df-sum 15714 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cn 23287 df-cnp 23288 df-cmp 23447 df-tx 23622 df-hmeo 23815 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-ovol 25526 df-vol 25527 df-mbf 25681 df-itg1 25682 df-itg2 25683 df-ibl 25684 df-0p 25732

This theorem is referenced by: bddibl 25902 itgsubstlem 26110 3factsumint1 42638 fourierdlem16 46697 fourierdlem21 46702 fourierdlem22 46703 fourierdlem83 46763

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