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Theorem iblcnlem 24402
 Description: Expand out the universal quantifier in isibl2 24380. (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 24381 . . 3 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
21a1i 11 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn))
3 simp1 1133 . . 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 2798 . . . . . 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 2798 . . . . . 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 2798 . . . . . 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 2798 . . . . . 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 10625 . . . . . . . 8 0 ∈ ℂ
109elimel 4492 . . . . . . 7 if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
1110a1i 11 . . . . . 6 ((𝜑𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ)
125, 6, 7, 8, 11iblcnlem1 24401 . . . . 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 484 . . . 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 2798 . . . . . 6 𝐴 = 𝐴
15 mbff 24239 . . . . . . . . 9 ((𝑥𝐴𝐵) ∈ MblFn → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ)
16 eqid 2798 . . . . . . . . . . . 12 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
17 itgcnlem.v . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑉)
1816, 17dmmptd 6466 . . . . . . . . . . 11 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
1918feq2d 6474 . . . . . . . . . 10 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ))
2019biimpa 480 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2115, 20sylan2 595 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2216fmpt 6852 . . . . . . . 8 (∀𝑥𝐴 𝐵 ∈ ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ)
2321, 22sylibr 237 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥𝐴 𝐵 ∈ ℂ)
24 iftrue 4431 . . . . . . . 8 (𝐵 ∈ ℂ → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
2524ralimi 3128 . . . . . . 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 5118 . . . . . 6 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵) → (𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥𝐴𝐵))
2814, 26, 27sylancr 590 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥𝐴𝐵))
2928eleq1d 2874 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿1 ↔ (𝑥𝐴𝐵) ∈ 𝐿1))
3028eleq1d 2874 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
31 eqid 2798 . . . . . . . . . 10 ℝ = ℝ
3224imim2i 16 . . . . . . . . . . . . . . . 16 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥𝐴 → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵))
3332imp 410 . . . . . . . . . . . . . . 15 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
3433fveq2d 6650 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℜ‘𝐵))
3534ibllem 24378 . . . . . . . . . . . . 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 3125 . . . . . . . . . . 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 5118 . . . . . . . . . 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 590 . . . . . . . . 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 6650 . . . . . . . 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, 42eqtr4di 2851 . . . . . . 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 2874 . . . . . 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 10872 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℜ‘𝐵))
4645ibllem 24378 . . . . . . . . . . . . 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 3125 . . . . . . . . . . 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 5118 . . . . . . . . . 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 590 . . . . . . . . 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 6650 . . . . . . . 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, 53eqtr4di 2851 . . . . . . 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 2874 . . . . . 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 633 . . . . 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 6650 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℑ‘𝐵))
5857ibllem 24378 . . . . . . . . . . . . 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 3125 . . . . . . . . . . 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 5118 . . . . . . . . . 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 590 . . . . . . . . 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 6650 . . . . . . . 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, 65eqtr4di 2851 . . . . . . 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 2874 . . . . . 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 10872 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℑ‘𝐵))
6968ibllem 24378 . . . . . . . . . . . . 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 3125 . . . . . . . . . . 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 5118 . . . . . . . . . 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 590 . . . . . . . . 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 6650 . . . . . . . 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, 76eqtr4di 2851 . . . . . . 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 2874 . . . . . 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 633 . . . . 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 1433 . . . 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 312 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
8281ex 416 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))))
832, 4, 82pm5.21ndd 384 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ifcif 4425   class class class wbr 5031   ↦ cmpt 5111  dom cdm 5520  ⟶wf 6321  ‘cfv 6325  ℂcc 10527  ℝcr 10528  0cc0 10529   ≤ cle 10668  -cneg 10863  ℜcre 14451  ℑcim 14452  MblFncmbf 24228  ∫2citg2 24230  𝐿1cibl 24231 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 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 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 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-2 11691  df-3 11692  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889  df-seq 13368  df-exp 13429  df-cj 14453  df-re 14454  df-im 14455  df-mbf 24233  df-ibl 24236 This theorem is referenced by:  itgcnlem  24403  iblrelem  24404  ibladd  24434  ibladdnc  35133
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