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Theorem bddmulibl 25723

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 25509	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
2	1	ad2antrr 723	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹:dom 𝐹⟶ℂ)
3	2	ffnd 6712	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 Fn dom 𝐹)
4		iblmbf 25652	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
5	4	ad2antlr 724	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ MblFn)
6		mbff 25509	. . . . . . 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 6712	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 Fn dom 𝐺)
9		mbfdm 25510	. . . . . 6 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
10	9	ad2antrr 723	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐹 ∈ dom vol)
11		mbfdm 25510	. . . . . 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 2726	. . . . 5 ⊢ (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14		eqidd 2727	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
15		eqidd 2727	. . . . 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 7676	. . . 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 7440	. . . . 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 764	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 ∈ MblFn)
19	18, 5	mbfmul 25611	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ MblFn)
20	16, 19	eqeltrrd 2828	. . . . 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 15290	. . . . . . . . 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 25520	. . . . . . . 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 7126	. . . . . . 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 7110	. . . . . . . 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 11169	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
27		ssid 3999	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
28		cncfss 24774	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
29	26, 27, 28	mp2an 689	. . . . . . . . . 10 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
30		abscncf 24776	. . . . . . . . . 10 ⊢ abs ∈ (ℂ–cn→ℝ)
31	29, 30	sselii 3974	. . . . . . . . 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 25542	. . . . . . . 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 1368	. . . . . . 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 2828	. . . . . 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 15389	. . . . . . . . . . . 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 11268	. . . . . . . . . . 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 15397	. . . . . . . . . . 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 13440	. . . . . . . . . . 11 ⊢ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))))
40	37, 38, 39	sylanbrc 582	. . . . . . . . . 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 13442	. . . . . . . . . . 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 4558	. . . . . . . . 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 7110	. . . . . . 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 11203	. . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℝ)
49	48	ad2antrr 723	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
50		elinel2 4191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐺)
51		ffvelcdm 7077	. . . . . . . . . . . . . . . . . . 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 15389	. . . . . . . . . . . . . . . . 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 15397	. . . . . . . . . . . . . . . . 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 13437	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘(𝐺‘𝑧)) ∈ (0[,)+∞) ↔ ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
56	53, 54, 55	sylanbrc 582	. . . . . . . . . . . . . . . 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 13441	. . . . . . . . . . . . . . . . 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 4558	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . 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 5731	. . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . 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 7687	. . . . . . . . . . . . 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 7503	. . . . . . . . . . . . . . 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 11246	. . . . . . . . . . . . . . . . . 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 11417	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · 0) = 0)
69	68	ifeq2d 4543	. . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . 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 5243	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 6889	. . . . . . . . . . 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 7110	. . . . . . . . . . . . 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 4224	. . . . . . . . . . . . . . . . . 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 25521	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
80	7	ffvelcdmda 7080	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
81	7	feqmptd 6954	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 = (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)))
82		simplr 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ 𝐿¹)
83	81, 82	eqeltrrd 2828	. . . . . . . . . . . . . . . . 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 25689	. . . . . . . . . . . . . . . 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 25713	. . . . . . . . . . . . . . 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 25677	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 774	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℝ)
91		neq0 4340	. . . . . . . . . . . . . . 15 ⊢ (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
92		0re 11220	. . . . . . . . . . . . . . . . . . 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 4190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐹)
95		ffvelcdm 7077	. . . . . . . . . . . . . . . . . . . 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 15389	. . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 𝑥 ∈ ℝ)
99	96	absge0d 15397	. . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)
101		2fveq3 6890	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑧)))
102	101	breq1d 5151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → ((abs‘(𝐹‘𝑦)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑧)) ≤ 𝑥))
103	102	rspccva 3605	. . . . . . . . . . . . . . . . . . 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 11375	. . . . . . . . . . . . . . . . 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 1928	. . . . . . . . . . . . . . 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 241	. . . . . . . . . . . . . 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 13437	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
111	90, 109, 110	sylanbrc 582	. . . . . . . . . . . 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 25632	. . . . . . . . . . 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 2768	. . . . . . . . . 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 11248	. . . . . . . . . 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 2827	. . . . . . . . 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 4325	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑧 ∈ ∅
118		eleq2 2816	. . . . . . . . . . . . . 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 4532	. . . . . . . . . . . . 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 5243	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ 0))
123		fconstmpt 5731	. . . . . . . . . . 11 ⊢ (ℝ × {0}) = (𝑧 ∈ ℝ ↦ 0)
124	122, 123	eqtr4di 2784	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (ℝ × {0}))
125	124	fveq2d 6889	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (∫₂‘(ℝ × {0})))
126		itg20 25622	. . . . . . . . . 10 ⊢ (∫₂‘(ℝ × {0})) = 0
127	126, 92	eqeltri 2823	. . . . . . . . 9 ⊢ (∫₂‘(ℝ × {0})) ∈ ℝ
128	125, 127	eqeltrdi 2835	. . . . . . . 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 11248	. . . . . . . . . . . . 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 11268	. . . . . . . . . . . 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 11795	. . . . . . . . . . . 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 13440	. . . . . . . . . . . 12 ⊢ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (𝑥 · (abs‘(𝐺‘𝑧)))))
134	131, 132, 133	sylanbrc 582	. . . . . . . . . . 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 4558	. . . . . . . . . 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 7110	. . . . . . . 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 15407	. . . . . . . . . . . . 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 15231	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → (abs‘(𝐺‘𝑧)) ∈ ℝ)
140		absge0 15240	. . . . . . . . . . . . . . . 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 12072	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 5163	. . . . . . . . . . . 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 4529	. . . . . . . . . . . . 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 4529	. . . . . . . . . . . . 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 5173	. . . . . . . . . . 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 12317	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
152	151	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 0)
153		iffalse 4532	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = 0)
154	152, 153, 120	3brtr4d 5173	. . . . . . . . . . . 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 810	. . . . . . . . . 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 3144	. . . . . . . . 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 2727	. . . . . . . . . 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 2727	. . . . . . . . . 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 7688	. . . . . . . . 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 25624	. . . . . . . 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 1368	. . . . . . 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 25623	. . . . . . 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 1368	. . . . . 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 25677	. . . . . 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 710	. . . . 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 25714	. . . 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 2827	. . 3 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹)
171	170	rexlimdvaa 3150	. 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹))
172	171	3impia 1114	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 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 ifcif 4523 {csn 4623 class class class wbr 5141 ↦ cmpt 5224 × cxp 5667 dom cdm 5669 ∘ ccom 5673 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 ∘_f cof 7665 ∘_r cofr 7666 ℂcc 11110 ℝcr 11111 0cc0 11112 · cmul 11117 +∞cpnf 11249 ℝ^*cxr 11251 ≤ cle 11253 [,)cico 13332 [,]cicc 13333 abscabs 15187 –cn→ccncf 24751 volcvol 25347 MblFncmbf 25498 ∫₂citg2 25500 𝐿¹cibl 25501

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cc 10432 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-ovol 25348 df-vol 25349 df-mbf 25503 df-itg1 25504 df-itg2 25505 df-ibl 25506 df-0p 25554

This theorem is referenced by: bddibl 25724 itgsubstlem 25938 3factsumint1 41402 fourierdlem16 45411 fourierdlem21 45416 fourierdlem22 45417 fourierdlem83 45477

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