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Theorem iblcnlem 25153
Description: Expand out the universal quantifier in isibl2 25131. (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 25132 . . 3 ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn)
21a1i 11 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ 𝐿1 → (𝑥𝐴𝐵) ∈ MblFn))
3 simp1 1136 . . 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 2736 . . . . . 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 2736 . . . . . 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 2736 . . . . . 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 2736 . . . . . 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 11147 . . . . . . . 8 0 ∈ ℂ
109elimel 4555 . . . . . . 7 if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ
1110a1i 11 . . . . . 6 ((𝜑𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) ∈ ℂ)
125, 6, 7, 8, 11iblcnlem1 25152 . . . . 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 481 . . . 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 2736 . . . . . 6 𝐴 = 𝐴
15 mbff 24989 . . . . . . . . 9 ((𝑥𝐴𝐵) ∈ MblFn → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ)
16 eqid 2736 . . . . . . . . . . . 12 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
17 itgcnlem.v . . . . . . . . . . . 12 ((𝜑𝑥𝐴) → 𝐵𝑉)
1816, 17dmmptd 6646 . . . . . . . . . . 11 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
1918feq2d 6654 . . . . . . . . . 10 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ))
2019biimpa 477 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℂ) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2115, 20sylan2 593 . . . . . . . 8 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → (𝑥𝐴𝐵):𝐴⟶ℂ)
2216fmpt 7058 . . . . . . . 8 (∀𝑥𝐴 𝐵 ∈ ℂ ↔ (𝑥𝐴𝐵):𝐴⟶ℂ)
2321, 22sylibr 233 . . . . . . 7 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ∀𝑥𝐴 𝐵 ∈ ℂ)
24 iftrue 4492 . . . . . . . 8 (𝐵 ∈ ℂ → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
2524ralimi 3086 . . . . . . 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 5197 . . . . . 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 2822 . . . 4 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ 𝐿1 ↔ (𝑥𝐴𝐵) ∈ 𝐿1))
3028eleq1d 2822 . . . . 5 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴 ↦ if(𝐵 ∈ ℂ, 𝐵, 0)) ∈ MblFn ↔ (𝑥𝐴𝐵) ∈ MblFn))
31 eqid 2736 . . . . . . . . . 10 ℝ = ℝ
3224imim2i 16 . . . . . . . . . . . . . . . 16 ((𝑥𝐴𝐵 ∈ ℂ) → (𝑥𝐴 → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵))
3332imp 407 . . . . . . . . . . . . . . 15 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → if(𝐵 ∈ ℂ, 𝐵, 0) = 𝐵)
3433fveq2d 6846 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℜ‘𝐵))
3534ibllem 25129 . . . . . . . . . . . . 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 3081 . . . . . . . . . . 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 5197 . . . . . . . . . 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 6846 . . . . . . . 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 2794 . . . . . . 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 2822 . . . . . 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 11395 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℜ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℜ‘𝐵))
4645ibllem 25129 . . . . . . . . . . . . 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 3081 . . . . . . . . . . 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 5197 . . . . . . . . . 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 6846 . . . . . . . 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 2794 . . . . . . 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 2822 . . . . . 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 631 . . . . 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 6846 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → (ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = (ℑ‘𝐵))
5857ibllem 25129 . . . . . . . . . . . . 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 3081 . . . . . . . . . . 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 5197 . . . . . . . . . 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 6846 . . . . . . . 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 2794 . . . . . . 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 2822 . . . . . 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 11395 . . . . . . . . . . . . . 14 (((𝑥𝐴𝐵 ∈ ℂ) ∧ 𝑥𝐴) → -(ℑ‘if(𝐵 ∈ ℂ, 𝐵, 0)) = -(ℑ‘𝐵))
6968ibllem 25129 . . . . . . . . . . . . 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 3081 . . . . . . . . . . 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 5197 . . . . . . . . . 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 6846 . . . . . . . 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 2794 . . . . . . 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 2822 . . . . . 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 631 . . . . 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 1436 . . . 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 308 . . 3 ((𝜑 ∧ (𝑥𝐴𝐵) ∈ MblFn) → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ))))
8281ex 413 . 2 (𝜑 → ((𝑥𝐴𝐵) ∈ MblFn → ((𝑥𝐴𝐵) ∈ 𝐿1 ↔ ((𝑥𝐴𝐵) ∈ MblFn ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ) ∧ (𝑇 ∈ ℝ ∧ 𝑈 ∈ ℝ)))))
832, 4, 82pm5.21ndd 380 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  ifcif 4486   class class class wbr 5105  cmpt 5188  dom cdm 5633  wf 6492  cfv 6496  cc 11049  cr 11050  0cc0 11051  cle 11190  -cneg 11386  cre 14982  cim 14983  MblFncmbf 24978  2citg2 24980  𝐿1cibl 24981
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-mbf 24983  df-ibl 24986
This theorem is referenced by:  itgcnlem  25154  iblrelem  25155  ibladd  25185  ibladdnc  36135
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