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Theorem bddmulibl 26168

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 25955	. . . . . . 7 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)
2	1	ad2antrr 739	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹:dom 𝐹⟶ℂ)
3	2	ffnd 6715	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 Fn dom 𝐹)
4		iblmbf 26097	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐿¹ → 𝐺 ∈ MblFn)
5	4	ad2antlr 740	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ MblFn)
6		mbff 25955	. . . . . . 7 ⊢ (𝐺 ∈ MblFn → 𝐺:dom 𝐺⟶ℂ)
7	5, 6	syl 18	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺:dom 𝐺⟶ℂ)
8	7	ffnd 6715	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 Fn dom 𝐺)
9		mbfdm 25956	. . . . . 6 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol)
10	9	ad2antrr 739	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐹 ∈ dom vol)
11		mbfdm 25956	. . . . . 6 ⊢ (𝐺 ∈ MblFn → dom 𝐺 ∈ dom vol)
12	5, 11	syl 18	. . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐺 ∈ dom vol)
13		eqid 2760	. . . . 5 ⊢ (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14		eqidd 2761	. . . . 5 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧))
15		eqidd 2761	. . . . 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 7706	. . . 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 7459	. . . . 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 779	. . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 ∈ MblFn)
19	18, 5	mbfmul 26056	. . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘_f · 𝐺) ∈ MblFn)
20	16, 19	eqeltrrd 2861	. . . . 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 15514	. . . . . . . . 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 25966	. . . . . . . 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 7138	. . . . . . 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 7120	. . . . . . . 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 11264	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
27		ssid 3931	. . . . . . . . . . 11 ⊢ ℂ ⊆ ℂ
28		cncfss 25229	. . . . . . . . . . 11 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ))
29	26, 27, 28	mp2an 705	. . . . . . . . . 10 ⊢ (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)
30		abscncf 25231	. . . . . . . . . 10 ⊢ abs ∈ (ℂ–cn→ℝ)
31	29, 30	sselii 3906	. . . . . . . . 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 25988	. . . . . . . 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 1398	. . . . . . 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 2861	. . . . . 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 15615	. . . . . . . . . . . 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 11366	. . . . . . . . . . 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 15623	. . . . . . . . . . 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 13595	. . . . . . . . . . 11 ⊢ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))))
40	37, 38, 39	sylanbrc 595	. . . . . . . . . 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 13597	. . . . . . . . . . 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 4517	. . . . . . . . 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 486	. . . . . . . 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 7120	. . . . . . 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 11298	. . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . 15 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℝ)
49	48	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ)
50		elinel2 4098	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐺)
51		ffvelcdm 7086	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
52	7, 50, 51	syl2an 608	. . . . . . . . . . . . . . . . . 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 15615	. . . . . . . . . . . . . . . . 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 15623	. . . . . . . . . . . . . . . . 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 13592	. . . . . . . . . . . . . . . . 17 ⊢ ((abs‘(𝐺‘𝑧)) ∈ (0[,)+∞) ↔ ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
56	53, 54, 55	sylanbrc 595	. . . . . . . . . . . . . . . 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 13596	. . . . . . . . . . . . . . . . 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 4517	. . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . 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 5713	. . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . 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 7717	. . . . . . . . . . . . 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 7523	. . . . . . . . . . . . . . 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 11344	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℂ)
67	66	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℂ)
68	67	mul01d 11516	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · 0) = 0)
69	68	ifeq2d 4502	. . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . 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 5198	. . . . . . . . . . . . 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 2795	. . . . . . . . . . . 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 6894	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 7120	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . . . . 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 25967	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
80	7	ffvelcdmda 7089	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ)
81	7	feqmptd 6958	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 = (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)))
82		simplr 781	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ 𝐿¹)
83	81, 82	eqeltrrd 2861	. . . . . . . . . . . . . . . . 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 26134	. . . . . . . . . . . . . . . 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 26158	. . . . . . . . . . . . . . 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 26122	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 501	. . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 789	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℝ)
91		neq0 4298	. . . . . . . . . . . . . . 15 ⊢ (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
92		0re 11317	. . . . . . . . . . . . . . . . . . 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 4097	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐹)
95		ffvelcdm 7086	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ)
96	2, 94, 95	syl2an 608	. . . . . . . . . . . . . . . . . . 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 15615	. . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 𝑥 ∈ ℝ)
99	96	absge0d 15623	. . . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)
101		2fveq3 6895	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑧 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑧)))
102	101	breq1d 5112	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → ((abs‘(𝐹‘𝑦)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑧)) ≤ 𝑥))
103	102	rspccva 3575	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 ∧ 𝑧 ∈ dom 𝐹) → (abs‘(𝐹‘𝑧)) ≤ 𝑥)
104	100, 94, 103	syl2an 608	. . . . . . . . . . . . . . . . . 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 11474	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ 𝑥)
106	105	ex 418	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥))
107	106	exlimdv 1965	. . . . . . . . . . . . . . 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 245	. . . . . . . . . . . . . 14 ⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ → 0 ≤ 𝑥))
109	108	imp 412	. . . . . . . . . . . . 13 ⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 0 ≤ 𝑥)
110		elrege0 13592	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
111	90, 109, 110	sylanbrc 595	. . . . . . . . . . . 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 26077	. . . . . . . . . . 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 2797	. . . . . . . . . 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 11346	. . . . . . . . . 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 2860	. . . . . . . . 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 418	. . . . . . . 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 4282	. . . . . . . . . . . . . 14 ⊢ ¬ 𝑧 ∈ ∅
118		eleq2 2849	. . . . . . . . . . . . . 14 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↔ 𝑧 ∈ ∅))
119	117, 118	mtbiri 330	. . . . . . . . . . . . 13 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺))
120		iffalse 4490	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = 0)
121	119, 120	syl 18	. . . . . . . . . . . 12 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = 0)
122	121	mpteq2dv 5198	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ 0))
123		fconstmpt 5713	. . . . . . . . . . 11 ⊢ (ℝ × {0}) = (𝑧 ∈ ℝ ↦ 0)
124	122, 123	eqtr4di 2813	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (ℝ × {0}))
125	124	fveq2d 6894	. . . . . . . . 9 ⊢ ((dom 𝐹 ∩ dom 𝐺) = ∅ → (∫₂‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (∫₂‘(ℝ × {0})))
126		itg20 26067	. . . . . . . . . 10 ⊢ (∫₂‘(ℝ × {0})) = 0
127	126, 92	eqeltri 2856	. . . . . . . . 9 ⊢ (∫₂‘(ℝ × {0})) ∈ ℝ
128	125, 127	eqeltrdi 2868	. . . . . . . 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 183	. . . . . . 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 11346	. . . . . . . . . . . . 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 11366	. . . . . . . . . . . 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 11900	. . . . . . . . . . . 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 13595	. . . . . . . . . . . 12 ⊢ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ^* ∧ 0 ≤ (𝑥 · (abs‘(𝐺‘𝑧)))))
134	131, 132, 133	sylanbrc 595	. . . . . . . . . . 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 4517	. . . . . . . . . 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 486	. . . . . . . . 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 7120	. . . . . . . 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 15633	. . . . . . . . . . . . 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 15454	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → (abs‘(𝐺‘𝑧)) ∈ ℝ)
140		absge0 15463	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑧) ∈ ℂ → 0 ≤ (abs‘(𝐺‘𝑧)))
141	139, 140	jca 521	. . . . . . . . . . . . . . 15 ⊢ ((𝐺‘𝑧) ∈ ℂ → ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤ (abs‘(𝐺‘𝑧))))
142	52, 141	syl 18	. . . . . . . . . . . . . 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 12178	. . . . . . . . . . . . . 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 1400	. . . . . . . . . . . . 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 5126	. . . . . . . . . . . 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 4487	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 4487	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧))))
149	148	adantl 487	. . . . . . . . . . . 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 5136	. . . . . . . . . . 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 12451	. . . . . . . . . . . . . 14 ⊢ 0 ≤ 0
152	151	a1i 11	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 0)
153		iffalse 4490	. . . . . . . . . . . . 13 ⊢ (¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = 0)
154	152, 153, 120	3brtr4d 5136	. . . . . . . . . . . 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 487	. . . . . . . . . . 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 825	. . . . . . . . . 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 2761	. . . . . . . . . 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 2761	. . . . . . . . . 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 7718	. . . . . . . . 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 260	. . . . . . . 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 26069	. . . . . . . 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 1398	. . . . . . 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 26068	. . . . . . 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 1398	. . . . . 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 26122	. . . . . 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 726	. . . . 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 26159	. . . 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 2860	. . 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 1135	1 ⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿¹ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘_f · 𝐺) ∈ 𝐿¹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ⊆ wss 3896 ∩ cin 4088 ∅c0 4277 ifcif 4481 {csn 4584 class class class wbr 5102 ↦ cmpt 5185 × cxp 5649 dom cdm 5651 ∘ ccom 5655 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ∘_f cof 7695 ∘_r cofr 7696 ℂcc 11205 ℝcr 11206 0cc0 11207 · cmul 11212 +∞cpnf 11347 ℝ^*cxr 11349 ≤ cle 11351 [,)cico 13485 [,]cicc 13486 abscabs 15410 –cn→ccncf 25206 volcvol 25793 MblFncmbf 25944 ∫₂citg2 25946 𝐿¹cibl 25947

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cc 10520 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-ofr 7698 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-oadd 8488 df-omul 8489 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9511 df-dju 9989 df-card 10027 df-acn 10030 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-q 13083 df-rp 13128 df-xneg 13248 df-xadd 13249 df-xmul 13250 df-ioo 13487 df-ioc 13488 df-ico 13489 df-icc 13490 df-fz 13647 df-fzo 13796 df-fl 13939 df-mod 14017 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-rest 17602 df-topn 17603 df-0g 17621 df-gsum 17622 df-topgen 17623 df-pt 17624 df-prds 17627 df-xrs 17683 df-qtop 17688 df-imas 17689 df-xps 17691 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-submnd 18988 df-mulg 19287 df-cntz 19540 df-cmn 20005 df-psmet 21679 df-xmet 21680 df-met 21681 df-bl 21682 df-mopn 21683 df-cnfld 21688 df-top 23221 df-topon 23238 df-topsp 23260 df-bases 23273 df-cn 23554 df-cnp 23555 df-cmp 23714 df-tx 23890 df-hmeo 24083 df-xms 24648 df-ms 24649 df-tms 24650 df-cncf 25208 df-ovol 25794 df-vol 25795 df-mbf 25949 df-itg1 25950 df-itg2 25951 df-ibl 25952 df-0p 26000

This theorem is used by: bddibl 26169 itgsubstlem 26377 3factsumint1 43208 fourierdlem16 47269 fourierdlem21 47274 fourierdlem22 47275 fourierdlem83 47335

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