| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | itg2add.f1 | . . 3
⊢ (𝜑 → 𝐹 ∈ MblFn) | 
| 2 |  | itg2add.f2 | . . 3
⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) | 
| 3 | 1, 2 | mbfi1fseq 25756 | . 2
⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥))) | 
| 4 |  | itg2add.g1 | . . 3
⊢ (𝜑 → 𝐺 ∈ MblFn) | 
| 5 |  | itg2add.g2 | . . 3
⊢ (𝜑 → 𝐺:ℝ⟶(0[,)+∞)) | 
| 6 | 4, 5 | mbfi1fseq 25756 | . 2
⊢ (𝜑 → ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))) | 
| 7 |  | exdistrv 1955 | . . 3
⊢
(∃𝑓∃𝑔((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))) ↔ (∃𝑓(𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) | 
| 8 | 1 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → 𝐹 ∈ MblFn) | 
| 9 | 2 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → 𝐹:ℝ⟶(0[,)+∞)) | 
| 10 |  | itg2add.f3 | . . . . . . 7
⊢ (𝜑 →
(∫2‘𝐹)
∈ ℝ) | 
| 11 | 10 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → (∫2‘𝐹) ∈
ℝ) | 
| 12 | 4 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → 𝐺 ∈ MblFn) | 
| 13 | 5 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → 𝐺:ℝ⟶(0[,)+∞)) | 
| 14 |  | itg2add.g3 | . . . . . . 7
⊢ (𝜑 →
(∫2‘𝐺)
∈ ℝ) | 
| 15 | 14 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → (∫2‘𝐺) ∈
ℝ) | 
| 16 |  | simprl1 1219 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → 𝑓:ℕ⟶dom
∫1) | 
| 17 |  | simprl2 1220 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → ∀𝑛 ∈ ℕ (0𝑝
∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1)))) | 
| 18 |  | simprl3 1221 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) | 
| 19 |  | simprr1 1222 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → 𝑔:ℕ⟶dom
∫1) | 
| 20 |  | simprr2 1223 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → ∀𝑛 ∈ ℕ (0𝑝
∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1)))) | 
| 21 |  | simprr3 1224 | . . . . . 6
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)) | 
| 22 | 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21 | itg2addlem 25793 | . . . . 5
⊢ ((𝜑 ∧ ((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥)))) → (∫2‘(𝐹 ∘f + 𝐺)) =
((∫2‘𝐹) + (∫2‘𝐺))) | 
| 23 | 22 | ex 412 | . . . 4
⊢ (𝜑 → (((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))) → (∫2‘(𝐹 ∘f + 𝐺)) =
((∫2‘𝐹) + (∫2‘𝐺)))) | 
| 24 | 23 | exlimdvv 1934 | . . 3
⊢ (𝜑 → (∃𝑓∃𝑔((𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ (𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))) → (∫2‘(𝐹 ∘f + 𝐺)) =
((∫2‘𝐹) + (∫2‘𝐺)))) | 
| 25 | 7, 24 | biimtrrid 243 | . 2
⊢ (𝜑 → ((∃𝑓(𝑓:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑓‘𝑛) ∧ (𝑓‘𝑛) ∘r ≤ (𝑓‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑓‘𝑛)‘𝑥)) ⇝ (𝐹‘𝑥)) ∧ ∃𝑔(𝑔:ℕ⟶dom ∫1 ∧
∀𝑛 ∈ ℕ
(0𝑝 ∘r ≤ (𝑔‘𝑛) ∧ (𝑔‘𝑛) ∘r ≤ (𝑔‘(𝑛 + 1))) ∧ ∀𝑥 ∈ ℝ (𝑛 ∈ ℕ ↦ ((𝑔‘𝑛)‘𝑥)) ⇝ (𝐺‘𝑥))) → (∫2‘(𝐹 ∘f + 𝐺)) =
((∫2‘𝐹) + (∫2‘𝐺)))) | 
| 26 | 3, 6, 25 | mp2and 699 | 1
⊢ (𝜑 →
(∫2‘(𝐹
∘f + 𝐺)) =
((∫2‘𝐹) + (∫2‘𝐺))) |