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Theorem iblcnlem 23275
Description: Expand out the forall in isibl2 23253. (Contributed by Mario Carneiro, 6-Aug-2014.)
Hypotheses
Ref Expression
itgcnlem.r 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
itgcnlem.s 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
itgcnlem.t 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
itgcnlem.u 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
itgcnlem.v ((𝜑𝑥𝐴) → 𝐵𝑉)
Assertion
Ref Expression
iblcnlem (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ 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
StepHypRef Expression
1 iblmbf 23254 . . 3 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
21a1i 11 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn))
3 simp1 1053 . . 3 (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)) → (𝑥𝐴𝐵) ∈ MblFn)
43a1i 11 . 2 (𝜑 → (((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)) → (𝑥𝐴𝐵) ∈ MblFn))
5 eqid 2606 . . . . . 6 (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
6 eqid 2606 . . . . . 6 (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
7 eqid 2606 . . . . . 6 (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
8 eqid 2606 . . . . . 6 (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)))
9 0cn 9885 . . . . . . . 8 0 ∈ ℂ
109elimel 4096 . . . . . . 7 if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
1110a1i 11 . . . . . 6 ((𝜑𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ)
125, 6, 7, 8, 11iblcnlem1 23274 . . . . 5 (𝜑 → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ∧ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ∧ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ))))
1312adantr 479 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿1 ↔ ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ∧ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ∧ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ))))
14 eqid 2606 . . . . . 6 𝐴 = 𝐴
15 mbff 23114 . . . . . . . . 9 ((𝑥𝐴𝐵) ∈ MblFn → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ)
16 eqid 2606 . . . . . . . . . . . 12 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
17 itgcnlem.v . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑉)
1816, 17dmmptd 5920 . . . . . . . . . . 11 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
1918feq2d 5927 . . . . . . . . . 10 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ))
2019biimpa 499 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2115, 20sylan2 489 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2216fmpt 6271 . . . . . . . 8 (∀𝑥𝐴 𝐵 ∈ ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ)
2321, 22sylibr 222 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥𝐴 𝐵 ∈ ℂ)
24 iftrue 4038 . . . . . . . 8 (𝐵 ∈ ℂ → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
2524ralimi 2932 . . . . . . 7 (∀𝑥𝐴 𝐵 ∈ ℂ → ∀𝑥𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
2623, 25syl 17 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
27 mpteq12 4655 . . . . . 6 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵) → (𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥𝐴𝐵))
2814, 26, 27sylancr 693 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥𝐴𝐵))
2928eleq1d 2668 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿1 ↔ (𝑥𝐴𝐵) ∈ 𝐿1))
3028eleq1d 2668 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
31 eqid 2606 . . . . . . . . . 10 ℝ = ℝ
3224imim2i 16 . . . . . . . . . . . . . . . 16 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥𝐴 → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵))
3332imp 443 . . . . . . . . . . . . . . 15 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
3433fveq2d 6089 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℜ‘𝐵))
3534ibllem 23251 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵 ∈ ℂ) → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
3635a1d 25 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
3736ralimi2 2929 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
3823, 37syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))
39 mpteq12 4655 . . . . . . . . . 10 ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
4031, 38, 39sylancr 693 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
4140fveq2d 6089 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0))))
42 itgcnlem.r . . . . . . . 8 𝑅 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘𝐵)), (ℜ‘𝐵), 0)))
4341, 42syl6eqr 2658 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑅)
4443eleq1d 2668 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑅 ∈ ℝ))
4534negeqd 10123 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℜ‘𝐵))
4645ibllem 23251 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵 ∈ ℂ) → if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))
4746a1d 25 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
4847ralimi2 2929 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))
4923, 48syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))
50 mpteq12 4655 . . . . . . . . . 10 ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
5131, 49, 50sylancr 693 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
5251fveq2d 6089 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0))))
53 itgcnlem.s . . . . . . . 8 𝑆 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘𝐵)), -(ℜ‘𝐵), 0)))
5452, 53syl6eqr 2658 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑆)
5554eleq1d 2668 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑆 ∈ ℝ))
5644, 55anbi12d 742 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ↔ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ)))
5733fveq2d 6089 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℑ‘𝐵))
5857ibllem 23251 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵 ∈ ℂ) → if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))
5958a1d 25 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
6059ralimi2 2929 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))
6123, 60syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))
62 mpteq12 4655 . . . . . . . . . 10 ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
6331, 61, 62sylancr 693 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
6463fveq2d 6089 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0))))
65 itgcnlem.t . . . . . . . 8 𝑇 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘𝐵)), (ℑ‘𝐵), 0)))
6664, 65syl6eqr 2658 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑇)
6766eleq1d 2668 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑇 ∈ ℝ))
6857negeqd 10123 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℑ‘𝐵))
6968ibllem 23251 . . . . . . . . . . . . 13 ((𝑥𝐴𝐵 ∈ ℂ) → if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))
7069a1d 25 . . . . . . . . . . . 12 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥 ∈ ℝ → if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
7170ralimi2 2929 . . . . . . . . . . 11 (∀𝑥𝐴 𝐵 ∈ ℂ → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))
7223, 71syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))
73 mpteq12 4655 . . . . . . . . . 10 ((ℝ = ℝ ∧ ∀𝑥 ∈ ℝ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0) = if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
7431, 72, 73sylancr 693 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0)) = (𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
7574fveq2d 6089 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0))))
76 itgcnlem.u . . . . . . . 8 𝑈 = (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘𝐵)), -(ℑ‘𝐵), 0)))
7775, 76syl6eqr 2658 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) = 𝑈)
7877eleq1d 2668 . . . . . 6 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ↔ 𝑈 ∈ ℝ))
7967, 78anbi12d 742 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ↔ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))
8030, 56, 793anbi123d 1390 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ∧ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ) ∧ ((∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ ∧ (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥𝐴 ∧ 0 ≤ -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0))), -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)), 0))) ∈ ℝ)) ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
8113, 29, 803bitr3d 296 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
8281ex 448 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))))
832, 4, 82pm5.21ndd 367 1 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wral 2892  ifcif 4032   class class class wbr 4574  cmpt 4634  dom cdm 5025  wf 5783  cfv 5787  cc 9787  cr 9788  0cc0 9789  cle 9928  -cneg 10115  cre 13628  cim 13629  MblFncmbf 23103  2citg2 23105  𝐿1cibl 23106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-er 7603  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-3 10924  df-n0 11137  df-z 11208  df-uz 11517  df-fz 12150  df-seq 12616  df-exp 12675  df-cj 13630  df-re 13631  df-im 13632  df-mbf 23108  df-ibl 23111
This theorem is referenced by:  itgcnlem  23276  iblrelem  23277  ibladd  23307  ibladdnc  32437
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