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Theorem iblcnlem 25839
Description: Expand out the universal quantifier in isibl2 25816. (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 25817 . . 3 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
21a1i 11 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn))
3 simp1 1135 . . 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 2735 . . . . . 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 2735 . . . . . 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 2735 . . . . . 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 2735 . . . . . 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 11251 . . . . . . . 8 0 ∈ ℂ
109elimel 4600 . . . . . . 7 if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
1110a1i 11 . . . . . 6 ((𝜑𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ)
125, 6, 7, 8, 11iblcnlem1 25838 . . . . 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 480 . . . 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 2735 . . . . . 6 𝐴 = 𝐴
15 mbff 25674 . . . . . . . . 9 ((𝑥𝐴𝐵) ∈ MblFn → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ)
16 eqid 2735 . . . . . . . . . . . 12 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
17 itgcnlem.v . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑉)
1816, 17dmmptd 6714 . . . . . . . . . . 11 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
1918feq2d 6723 . . . . . . . . . 10 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ))
2019biimpa 476 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2115, 20sylan2 593 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2216fmpt 7130 . . . . . . . 8 (∀𝑥𝐴 𝐵 ∈ ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ)
2321, 22sylibr 234 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥𝐴 𝐵 ∈ ℂ)
24 iftrue 4537 . . . . . . . 8 (𝐵 ∈ ℂ → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
2524ralimi 3081 . . . . . . 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 5240 . . . . . 6 ((𝐴 = 𝐴 ∧ ∀𝑥𝐴 if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵) → (𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥𝐴𝐵))
2814, 26, 27sylancr 587 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) = (𝑥𝐴𝐵))
2928eleq1d 2824 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿1 ↔ (𝑥𝐴𝐵) ∈ 𝐿1))
3028eleq1d 2824 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
31 eqid 2735 . . . . . . . . . 10 ℝ = ℝ
3224imim2i 16 . . . . . . . . . . . . . . . 16 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥𝐴 → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵))
3332imp 406 . . . . . . . . . . . . . . 15 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
3433fveq2d 6911 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℜ‘𝐵))
3534ibllem 25814 . . . . . . . . . . . . 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 3076 . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6911 . . . . . . . 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 2793 . . . . . . 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 2824 . . . . . 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 11500 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℜ‘𝐵))
4645ibllem 25814 . . . . . . . . . . . . 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 3076 . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6911 . . . . . . . 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 2793 . . . . . . 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 2824 . . . . . 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 632 . . . . 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 6911 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℑ‘𝐵))
5857ibllem 25814 . . . . . . . . . . . . 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 3076 . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6911 . . . . . . . 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 2793 . . . . . . 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 2824 . . . . . 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 11500 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℑ‘𝐵))
6968ibllem 25814 . . . . . . . . . . . . 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 3076 . . . . . . . . . . 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 5240 . . . . . . . . . 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 587 . . . . . . . . 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 6911 . . . . . . . 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 2793 . . . . . . 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 2824 . . . . . 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 632 . . . . 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 1435 . . . 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 309 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
8281ex 412 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))))
832, 4, 82pm5.21ndd 379 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  ifcif 4531   class class class wbr 5148  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  cc 11151  cr 11152  0cc0 11153  cle 11294  -cneg 11491  cre 15133  cim 15134  MblFncmbf 25663  2citg2 25665  𝐿1cibl 25666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-mbf 25668  df-ibl 25671
This theorem is referenced by:  itgcnlem  25840  iblrelem  25841  ibladd  25871  ibladdnc  37664
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