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Theorem bddmulibl 24122

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 23909	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
2	1	ad2antrr 722	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹:dom 𝐹⟶ℂ)
3	2	ffnd 6383	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 Fn dom 𝐹)
4		iblmbf 24051	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
5	4	ad2antlr 723	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ MblFn)
6		mbff 23909	. . . . . . 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 6383	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 Fn dom 𝐺)
9		mbfdm 23910	. . . . . 6 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
10	9	ad2antrr 722	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐹 ∈ dom vol)
11		mbfdm 23910	. . . . . 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 2795	. . . . 5 ⊢ (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14		eqidd 2796	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
15		eqidd 2796	. . . . 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 7274	. . . 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 7050	. . . . 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 763	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 ∈ MblFn)
19	18, 5	mbfmul 24010	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_𝑓 · 𝐺) ∈ MblFn)
20	16, 19	eqeltrrd 2884	. . . . 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 14531	. . . . . . . . 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 23920	. . . . . . . 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 6757	. . . . . . 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 6742	. . . . . . . 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 10440	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
27		ssid 3910	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
28		cncfss 23190	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
29	26, 27, 28	mp2an 688	. . . . . . . . . 10 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
30		abscncf 23192	. . . . . . . . . 10 ⊢ abs ∈ (ℂ–cn→ℝ)
31	29, 30	sselii 3886	. . . . . . . . 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 23942	. . . . . . . 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 1364	. . . . . . 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 2884	. . . . . 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 14630	. . . . . . . . . . . 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 10537	. . . . . . . . . . 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 14638	. . . . . . . . . . 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 12695	. . . . . . . . . . 11 ⊢ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))))
40	37, 38, 39	sylanbrc 583	. . . . . . . . . 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 12697	. . . . . . . . . . 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 4415	. . . . . . . . 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 481	. . . . . . . 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 6742	. . . . . . 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 10474	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℝ)
49	48	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
50		elinel2 4094	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐺)
51		ffvelrn 6714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
52	7, 50, 51	syl2an 595	. . . . . . . . . . . . . . . . . 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 14630	. . . . . . . . . . . . . . . . 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 14638	. . . . . . . . . . . . . . . . 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 12692	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘(𝐺‘𝑧)) ∈ (0[,)+∞) ↔ ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
56	53, 54, 55	sylanbrc 583	. . . . . . . . . . . . . . . 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 12696	. . . . . . . . . . . . . . . . 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 4415	. . . . . . . . . . . . . . 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 722	. . . . . . . . . . . . . 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 5500	. . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . 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 7284	. . . . . . . . . . . . 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 7108	. . . . . . . . . . . . . . 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 10515	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℂ)
67	66	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℂ)
68	67	mul01d 10686	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · 0) = 0)
69	68	ifeq2d 4400	. . . . . . . . . . . . . . 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	syl5eq 2843	. . . . . . . . . . . . . 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 5056	. . . . . . . . . . . . 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 2831	. . . . . . . . . . . 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 6542	. . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 6742	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 4126	. . . . . . . . . . . . . . . . . 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 23921	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
80	7	ffvelrnda 6716	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
81	7	feqmptd 6601	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 = (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)))
82		simplr 765	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ 𝐿¹)
83	81, 82	eqeltrrd 2884	. . . . . . . . . . . . . . . . 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 24088	. . . . . . . . . . . . . . . 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 24112	. . . . . . . . . . . . . . 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 24076	. . . . . . . . . . . . . . 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 233	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 773	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℝ)
91		neq0 4229	. . . . . . . . . . . . . . 15 ⊢ (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
92		0re 10489	. . . . . . . . . . . . . . . . . . 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 4093	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐹)
95		ffvelrn 6714	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
96	2, 94, 95	syl2an 595	. . . . . . . . . . . . . . . . . . 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 14630	. . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 𝑥 ∈ ℝ)
99	96	absge0d 14638	. . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)
101		2fveq3 6543	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑧)))
102	101	breq1d 4972	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → ((abs‘(𝐹‘𝑦)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑧)) ≤ 𝑥))
103	102	rspccva 3558	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 ∧ 𝑧 ∈ dom 𝐹) → (abs‘(𝐹‘𝑧)) ≤ 𝑥)
104	100, 94, 103	syl2an 595	. . . . . . . . . . . . . . . . . 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 10644	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ 𝑥)
106	105	ex 413	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥))
107	106	exlimdv 1911	. . . . . . . . . . . . . . 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	syl5bi 243	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ → 0 ≤ 𝑥))
109	108	imp 407	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 0 ≤ 𝑥)
110		elrege0 12692	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
111	90, 109, 110	sylanbrc 583	. . . . . . . . . . . 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 24031	. . . . . . . . . . 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 2833	. . . . . . . . . 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 10517	. . . . . . . . . 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 2883	. . . . . . . . 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 413	. . . . . . . 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 4216	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑧 ∈ ∅
118		eleq2 2871	. . . . . . . . . . . . . 14 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↔ 𝑧 ∈ ∅))
119	117, 118	mtbiri 328	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
120		iffalse 4390	. . . . . . . . . . . . 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 5056	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ 0))
123		fconstmpt 5500	. . . . . . . . . . 11 ⊢ (ℝ × {0}) = (𝑧 ∈ ℝ ↦ 0)
124	122, 123	syl6eqr 2849	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (ℝ × {0}))
125	124	fveq2d 6542	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (∫₂‘(ℝ × {0})))
126		itg20 24021	. . . . . . . . . 10 ⊢ (∫₂‘(ℝ × {0})) = 0
127	126, 92	eqeltri 2879	. . . . . . . . 9 ⊢ (∫₂‘(ℝ × {0})) ∈ ℝ
128	125, 127	syl6eqel 2891	. . . . . . . 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 10517	. . . . . . . . . . . . 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 10537	. . . . . . . . . . . 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 11065	. . . . . . . . . . . 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 12695	. . . . . . . . . . . 12 ⊢ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (𝑥 · (abs‘(𝐺‘𝑧)))))
134	131, 132, 133	sylanbrc 583	. . . . . . . . . . 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 4415	. . . . . . . . . 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 481	. . . . . . . . 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 6742	. . . . . . . 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 14648	. . . . . . . . . . . . 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 14472	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → (abs‘(𝐺‘𝑧)) ∈ ℝ)
140		absge0 14481	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → 0 ≤ (abs‘(𝐺‘𝑧)))
141	139, 140	jca 512	. . . . . . . . . . . . . . 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 11342	. . . . . . . . . . . . . 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 1366	. . . . . . . . . . . . 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 4984	. . . . . . . . . . . 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 4387	. . . . . . . . . . . . 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 482	. . . . . . . . . . . 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 4387	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧))))
149	148	adantl 482	. . . . . . . . . . . 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 4994	. . . . . . . . . . 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 11586	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
152	151	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 0)
153		iffalse 4390	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = 0)
154	152, 153, 120	3brtr4d 4994	. . . . . . . . . . . 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 482	. . . . . . . . . . 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 809	. . . . . . . . . 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 3150	. . . . . . . . 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 2796	. . . . . . . . . 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 2796	. . . . . . . . . 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 7285	. . . . . . . . 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 258	. . . . . . . 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 24023	. . . . . . . 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 1364	. . . . . . 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 24022	. . . . . . 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 1364	. . . . . 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 24076	. . . . . 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 709	. . . . 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 24113	. . . 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 2883	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_𝑓 · 𝐺) ∈ 𝐿¹)
171	170	rexlimdvaa 3248	. 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 → (𝐹 ∘_𝑓 · 𝐺) ∈ 𝐿¹))
172	171	3impia 1110	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 396 ∧ w3a 1080 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ∀wral 3105 ∃wrex 3106 Vcvv 3437 ∩ cin 3858 ⊆ wss 3859 ∅c0 4211 ifcif 4381 {csn 4472 class class class wbr 4962 ↦ cmpt 5041 × cxp 5441 dom cdm 5443 ∘ ccom 5447 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 ∘_𝑓 cof 7265 ∘_𝑟 cofr 7266 ℂcc 10381 ℝcr 10382 0cc0 10383 · cmul 10388 +∞cpnf 10518 ℝ^*cxr 10520 ≤ cle 10522 [,)cico 12590 [,]cicc 12591 abscabs 14427 –cn→ccncf 23167 volcvol 23747 MblFncmbf 23898 ∫₂citg2 23900 𝐿¹cibl 23901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cc 9703 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-disj 4931 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-ofr 7268 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-omul 7958 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-dju 9176 df-card 9214 df-acn 9217 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ioc 12593 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cn 21519 df-cnp 21520 df-cmp 21679 df-tx 21854 df-hmeo 22047 df-xms 22613 df-ms 22614 df-tms 22615 df-cncf 23169 df-ovol 23748 df-vol 23749 df-mbf 23903 df-itg1 23904 df-itg2 23905 df-ibl 23906 df-0p 23954

This theorem is referenced by: bddibl 24123 itgsubstlem 24328 fourierdlem16 41950 fourierdlem21 41955 fourierdlem22 41956 fourierdlem83 42016

W3C validator