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Theorem bddmulibl 25794

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 25580	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
2	1	ad2antrr 726	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹:dom 𝐹⟶ℂ)
3	2	ffnd 6661	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 Fn dom 𝐹)
4		iblmbf 25722	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
5	4	ad2antlr 727	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ MblFn)
6		mbff 25580	. . . . . . 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 6661	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 Fn dom 𝐺)
9		mbfdm 25581	. . . . . 6 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
10	9	ad2antrr 726	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐹 ∈ dom vol)
11		mbfdm 25581	. . . . . 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 2734	. . . . 5 ⊢ (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14		eqidd 2735	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
15		eqidd 2735	. . . . 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 7629	. . . 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 7391	. . . . 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 766	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 ∈ MblFn)
19	18, 5	mbfmul 25681	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ MblFn)
20	16, 19	eqeltrrd 2835	. . . . 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 15259	. . . . . . . . 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 25591	. . . . . . . 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 7075	. . . . . . 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 7058	. . . . . . . 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 11081	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
27		ssid 3954	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
28		cncfss 24846	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
29	26, 27, 28	mp2an 692	. . . . . . . . . 10 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
30		abscncf 24848	. . . . . . . . . 10 ⊢ abs ∈ (ℂ–cn→ℝ)
31	29, 30	sselii 3928	. . . . . . . . 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 25613	. . . . . . . 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 1373	. . . . . . 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 2835	. . . . . 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 15360	. . . . . . . . . . . 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 11180	. . . . . . . . . . 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 15368	. . . . . . . . . . 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 13371	. . . . . . . . . . 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 13373	. . . . . . . . . . 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 4513	. . . . . . . . 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 480	. . . . . . . 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 7058	. . . . . . 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 11115	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℝ)
49	48	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
50		elinel2 4152	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐺)
51		ffvelcdm 7024	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
52	7, 50, 51	syl2an 596	. . . . . . . . . . . . . . . . . 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 15360	. . . . . . . . . . . . . . . . 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 15368	. . . . . . . . . . . . . . . . 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 13368	. . . . . . . . . . . . . . . . 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 13372	. . . . . . . . . . . . . . . . 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 4513	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 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 5684	. . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 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 7640	. . . . . . . . . . . . 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 7455	. . . . . . . . . . . . . . 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 11158	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℂ)
67	66	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℂ)
68	67	mul01d 11330	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · 0) = 0)
69	68	ifeq2d 4498	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 5190	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . 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 6836	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 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 7058	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 4188	. . . . . . . . . . . . . . . . . 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 25592	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
80	7	ffvelcdmda 7027	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
81	7	feqmptd 6900	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 = (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)))
82		simplr 768	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ 𝐿¹)
83	81, 82	eqeltrrd 2835	. . . . . . . . . . . . . . . . 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 25760	. . . . . . . . . . . . . . . 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 25784	. . . . . . . . . . . . . . 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 25748	. . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 776	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℝ)
91		neq0 4302	. . . . . . . . . . . . . . 15 ⊢ (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
92		0re 11132	. . . . . . . . . . . . . . . . . . 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 4151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐹)
95		ffvelcdm 7024	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
96	2, 94, 95	syl2an 596	. . . . . . . . . . . . . . . . . . 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 15360	. . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 𝑥 ∈ ℝ)
99	96	absge0d 15368	. . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)
101		2fveq3 6837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑧)))
102	101	breq1d 5106	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → ((abs‘(𝐹‘𝑦)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑧)) ≤ 𝑥))
103	102	rspccva 3573	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 ∧ 𝑧 ∈ dom 𝐹) → (abs‘(𝐹‘𝑧)) ≤ 𝑥)
104	100, 94, 103	syl2an 596	. . . . . . . . . . . . . . . . . 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 11288	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ 𝑥)
106	105	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥))
107	106	exlimdv 1934	. . . . . . . . . . . . . . 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 242	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ → 0 ≤ 𝑥))
109	108	imp 406	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 0 ≤ 𝑥)
110		elrege0 13368	. . . . . . . . . . . . 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 25702	. . . . . . . . . . 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 2771	. . . . . . . . . 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 11160	. . . . . . . . . 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 2834	. . . . . . . . 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 412	. . . . . . . 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 4288	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑧 ∈ ∅
118		eleq2 2823	. . . . . . . . . . . . . 14 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↔ 𝑧 ∈ ∅))
119	117, 118	mtbiri 327	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
120		iffalse 4486	. . . . . . . . . . . . 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 5190	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ 0))
123		fconstmpt 5684	. . . . . . . . . . 11 ⊢ (ℝ × {0}) = (𝑧 ∈ ℝ ↦ 0)
124	122, 123	eqtr4di 2787	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (ℝ × {0}))
125	124	fveq2d 6836	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (∫₂‘(ℝ × {0})))
126		itg20 25692	. . . . . . . . . 10 ⊢ (∫₂‘(ℝ × {0})) = 0
127	126, 92	eqeltri 2830	. . . . . . . . 9 ⊢ (∫₂‘(ℝ × {0})) ∈ ℝ
128	125, 127	eqeltrdi 2842	. . . . . . . 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 181	. . . . . . 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 11160	. . . . . . . . . . . . 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 11180	. . . . . . . . . . . 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 11712	. . . . . . . . . . . 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 13371	. . . . . . . . . . . 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 4513	. . . . . . . . . 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 480	. . . . . . . . 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 7058	. . . . . . . 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 15378	. . . . . . . . . . . . 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 15199	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → (abs‘(𝐺‘𝑧)) ∈ ℝ)
140		absge0 15208	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → 0 ≤ (abs‘(𝐺‘𝑧)))
141	139, 140	jca 511	. . . . . . . . . . . . . . 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 11993	. . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . 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 5118	. . . . . . . . . . . 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 4483	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 4483	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧))))
149	148	adantl 481	. . . . . . . . . . . 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 5128	. . . . . . . . . . 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 12244	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
152	151	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 0)
153		iffalse 4486	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = 0)
154	152, 153, 120	3brtr4d 5128	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 812	. . . . . . . . . 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 3130	. . . . . . . . 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 2735	. . . . . . . . . 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 2735	. . . . . . . . . 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 7641	. . . . . . . . 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 257	. . . . . . . 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 25694	. . . . . . . 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 1373	. . . . . . 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 25693	. . . . . . 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 1373	. . . . . 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 25748	. . . . . 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 713	. . . . 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 25785	. . . 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 2834	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹)
171	170	rexlimdvaa 3136	. 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹))
172	171	3impia 1117	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 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∀wral 3049 ∃wrex 3058 Vcvv 3438 ∩ cin 3898 ⊆ wss 3899 ∅c0 4283 ifcif 4477 {csn 4578 class class class wbr 5096 ↦ cmpt 5177 × cxp 5620 dom cdm 5622 ∘ ccom 5626 ⟶wf 6486 ‘cfv 6490 (class class class)co 7356 ∘_f cof 7618 ∘_r cofr 7619 ℂcc 11022 ℝcr 11023 0cc0 11024 · cmul 11029 +∞cpnf 11161 ℝ^*cxr 11163 ≤ cle 11165 [,)cico 13261 [,]cicc 13262 abscabs 15155 –cn→ccncf 24823 volcvol 25418 MblFncmbf 25569 ∫₂citg2 25571 𝐿¹cibl 25572

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-inf2 9548 ax-cc 10343 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-pre-sup 11102 ax-addf 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-disj 5064 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8763 df-pm 8764 df-ixp 8834 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-fsupp 9263 df-fi 9312 df-sup 9343 df-inf 9344 df-oi 9413 df-dju 9811 df-card 9849 df-acn 9852 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-div 11793 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-7 12211 df-8 12212 df-9 12213 df-n0 12400 df-z 12487 df-dec 12606 df-uz 12750 df-q 12860 df-rp 12904 df-xneg 13024 df-xadd 13025 df-xmul 13026 df-ioo 13263 df-ioc 13264 df-ico 13265 df-icc 13266 df-fz 13422 df-fzo 13569 df-fl 13710 df-mod 13788 df-seq 13923 df-exp 13983 df-hash 14252 df-cj 15020 df-re 15021 df-im 15022 df-sqrt 15156 df-abs 15157 df-limsup 15392 df-clim 15409 df-rlim 15410 df-sum 15608 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-starv 17190 df-sca 17191 df-vsca 17192 df-ip 17193 df-tset 17194 df-ple 17195 df-ds 17197 df-unif 17198 df-hom 17199 df-cco 17200 df-rest 17340 df-topn 17341 df-0g 17359 df-gsum 17360 df-topgen 17361 df-pt 17362 df-prds 17365 df-xrs 17421 df-qtop 17426 df-imas 17427 df-xps 17429 df-mre 17503 df-mrc 17504 df-acs 17506 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-mulg 18996 df-cntz 19244 df-cmn 19709 df-psmet 21299 df-xmet 21300 df-met 21301 df-bl 21302 df-mopn 21303 df-cnfld 21308 df-top 22836 df-topon 22853 df-topsp 22875 df-bases 22888 df-cn 23169 df-cnp 23170 df-cmp 23329 df-tx 23504 df-hmeo 23697 df-xms 24262 df-ms 24263 df-tms 24264 df-cncf 24825 df-ovol 25419 df-vol 25420 df-mbf 25574 df-itg1 25575 df-itg2 25576 df-ibl 25577 df-0p 25625

This theorem is referenced by: bddibl 25795 itgsubstlem 26009 3factsumint1 42214 fourierdlem16 46309 fourierdlem21 46314 fourierdlem22 46315 fourierdlem83 46375

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