iblcnlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > iblcnlem		Structured version Visualization version GIF version

Theorem iblcnlem 24392

Description: Expand out the universal quantifier in isibl2 24370. (Contributed by Mario Carneiro, 6-Aug-2014.)

**Hypotheses**
Ref	Expression
itgcnlem.r	⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
itgcnlem.s	⊢ 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
itgcnlem.t	⊢ 𝑇 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
itgcnlem.u	⊢ 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
itgcnlem.v	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)

**Assertion**
Ref	Expression
iblcnlem	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))

Distinct variable groups: 𝑥,𝐴 𝜑,𝑥 𝑥,𝑉 Allowed substitution hints: 𝐵(𝑥) 𝑅(𝑥) 𝑆(𝑥) 𝑇(𝑥) 𝑈(𝑥)

**Proof of Theorem iblcnlem**
Step	Hyp	Ref	Expression
1		iblmbf 24371	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
2	1	a1i 11	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn))
3		simp1 1133	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn)
4	3	a1i 11	. 2 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn))
5		eqid 2798	. . . . . 6 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
6		eqid 2798	. . . . . 6 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
7		eqid 2798	. . . . . 6 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
8		eqid 2798	. . . . . 6 ⊢ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
9		0cn 10622	. . . . . . . 8 ⊢ 0 ∈ ℂ
10	9	elimel 4492	. . . . . . 7 ⊢ if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ)
12	5, 6, 7, 8, 11	iblcnlem1 24391	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ∧ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ∧ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ))))
13	12	adantr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ∧ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ∧ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ))))
14		eqid 2798	. . . . . 6 ⊢ 𝐴 = 𝐴
15		mbff 24229	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn → (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ)
16		eqid 2798	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
17		itgcnlem.v	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)
18	16, 17	dmmptd 6465	. . . . . . . . . . 11 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴)
19	18	feq2d 6473	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ))
20	19	biimpa 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):dom (𝑥 ∈ 𝐴 ↦ 𝐵)⟶ℂ) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
21	15, 20	sylan2 595	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
22	16	fmpt 6851	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ)
23	21, 22	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ)
24		iftrue 4431	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
25	24	ralimi 3128	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ 𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
26	23, 25	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ∀𝑥 ∈ 𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
27		mpteq12 5117	. . . . . 6 ⊢ ((𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵) → (𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
28	14, 26, 27	sylancr 590	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥 ∈ 𝐴 ↦ 𝐵))
29	28	eleq1d 2874	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿¹ ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹))
30	28	eleq1d 2874	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn))
31		eqid 2798	. . . . . . . . . 10 ⊢ ℝ = ℝ
32	24	imim2i 16	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → (𝑥 ∈ 𝐴 → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵))
33	32	imp 410	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
34	33	fveq2d 6649	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℜ‘𝐵))
35	34	ibllem 24368	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
36	35	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
37	36	ralimi2 3125	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
38	23, 37	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
39		mpteq12 5117	. . . . . . . . . 10 ⊢ ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
40	31, 38, 39	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
41	40	fveq2d 6649	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))))
42		itgcnlem.r	. . . . . . . 8 ⊢ 𝑅 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
43	41, 42	eqtr4di 2851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑅)
44	43	eleq1d 2874	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑅 ∈ ℝ))
45	34	negeqd 10869	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℜ‘𝐵))
46	45	ibllem 24368	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))
47	46	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
48	47	ralimi2 3125	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))
49	23, 48	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))
50		mpteq12 5117	. . . . . . . . . 10 ⊢ ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
51	31, 49, 50	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
52	51	fveq2d 6649	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))))
53		itgcnlem.s	. . . . . . . 8 ⊢ 𝑆 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
54	52, 53	eqtr4di 2851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑆)
55	54	eleq1d 2874	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑆 ∈ ℝ))
56	44, 55	anbi12d 633	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ↔ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ)))
57	33	fveq2d 6649	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℑ‘𝐵))
58	57	ibllem 24368	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))
59	58	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
60	59	ralimi2 3125	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))
61	23, 60	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))
62		mpteq12 5117	. . . . . . . . . 10 ⊢ ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
63	31, 61, 62	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
64	63	fveq2d 6649	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))))
65		itgcnlem.t	. . . . . . . 8 ⊢ 𝑇 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
66	64, 65	eqtr4di 2851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑇)
67	66	eleq1d 2874	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑇 ∈ ℝ))
68	57	negeqd 10869	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) ∧ 𝑥 ∈ 𝐴) → -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℑ‘𝐵))
69	68	ibllem 24368	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))
70	69	a1d 25	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 → 𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
71	70	ralimi2 3125	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))
72	23, 71	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))
73		mpteq12 5117	. . . . . . . . . 10 ⊢ ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
74	31, 72, 73	sylancr 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
75	74	fveq2d 6649	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))))
76		itgcnlem.u	. . . . . . . 8 ⊢ 𝑈 = (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
77	75, 76	eqtr4di 2851	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑈)
78	77	eleq1d 2874	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑈 ∈ ℝ))
79	67, 78	anbi12d 633	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ↔ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
80	30, 56, 79	3anbi123d 1433	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → (((𝑥 ∈ 𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ∧ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ∧ ((∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫₂‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ)) ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
81	13, 29, 80	3bitr3d 312	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
82	81	ex 416	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))))
83	2, 4, 82	pm5.21ndd 384	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ 𝐿¹ ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 ℂcc 10524 ℝcr 10525 0cc0 10526 ≤ cle 10665 -cneg 10860 ℜcre 14448 ℑcim 14449 MblFncmbf 24218 ∫₂citg2 24220 𝐿¹cibl 24221

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-mbf 24223 df-ibl 24226

This theorem is referenced by: itgcnlem 24393 iblrelem 24394 ibladd 24424 ibladdnc 35114

W3C validator