| Step | Hyp | Ref
| Expression |
| 1 | | mbff 25660 |
. . . . . . 7
⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) |
| 2 | 1 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹:dom 𝐹⟶ℂ) |
| 3 | 2 | ffnd 6737 |
. . . . 5
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 Fn dom 𝐹) |
| 4 | | iblmbf 25802 |
. . . . . . . 8
⊢ (𝐺 ∈ 𝐿1
→ 𝐺 ∈
MblFn) |
| 5 | 4 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈ MblFn) |
| 6 | | mbff 25660 |
. . . . . . 7
⊢ (𝐺 ∈ MblFn → 𝐺:dom 𝐺⟶ℂ) |
| 7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺:dom 𝐺⟶ℂ) |
| 8 | 7 | ffnd 6737 |
. . . . 5
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 Fn dom 𝐺) |
| 9 | | mbfdm 25661 |
. . . . . 6
⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom
vol) |
| 10 | 9 | ad2antrr 726 |
. . . . 5
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐹 ∈ dom vol) |
| 11 | | mbfdm 25661 |
. . . . . 6
⊢ (𝐺 ∈ MblFn → dom 𝐺 ∈ dom
vol) |
| 12 | 5, 11 | syl 17 |
. . . . 5
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → dom 𝐺 ∈ dom vol) |
| 13 | | eqid 2737 |
. . . . 5
⊢ (dom
𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺) |
| 14 | | eqidd 2738 |
. . . . 5
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) = (𝐹‘𝑧)) |
| 15 | | eqidd 2738 |
. . . . 5
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) = (𝐺‘𝑧)) |
| 16 | 3, 8, 10, 12, 13, 14, 15 | offval 7706 |
. . . 4
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘f · 𝐺) = (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) |
| 17 | | ovexd 7466 |
. . . . 5
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ∈ V) |
| 18 | | simpll 767 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐹 ∈ MblFn) |
| 19 | 18, 5 | mbfmul 25761 |
. . . . . 6
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘f · 𝐺) ∈ MblFn) |
| 20 | 16, 19 | eqeltrrd 2842 |
. . . . 5
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ MblFn) |
| 21 | | absf 15376 |
. . . . . . . . 9
⊢
abs:ℂ⟶ℝ |
| 22 | 21 | a1i 11 |
. . . . . . . 8
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) →
abs:ℂ⟶ℝ) |
| 23 | 20, 17 | mbfmptcl 25671 |
. . . . . . . 8
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((𝐹‘𝑧) · (𝐺‘𝑧)) ∈ ℂ) |
| 24 | 22, 23 | cofmpt 7152 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (abs ∘ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) = (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))))) |
| 25 | 23 | fmpttd 7135 |
. . . . . . . 8
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))):(dom 𝐹 ∩ dom 𝐺)⟶ℂ) |
| 26 | | ax-resscn 11212 |
. . . . . . . . . . 11
⊢ ℝ
⊆ ℂ |
| 27 | | ssid 4006 |
. . . . . . . . . . 11
⊢ ℂ
⊆ ℂ |
| 28 | | cncfss 24925 |
. . . . . . . . . . 11
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) → (ℂ–cn→ℝ) ⊆ (ℂ–cn→ℂ)) |
| 29 | 26, 27, 28 | mp2an 692 |
. . . . . . . . . 10
⊢
(ℂ–cn→ℝ)
⊆ (ℂ–cn→ℂ) |
| 30 | | abscncf 24927 |
. . . . . . . . . 10
⊢ abs
∈ (ℂ–cn→ℝ) |
| 31 | 29, 30 | sselii 3980 |
. . . . . . . . 9
⊢ abs
∈ (ℂ–cn→ℂ) |
| 32 | 31 | a1i 11 |
. . . . . . . 8
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → abs ∈ (ℂ–cn→ℂ)) |
| 33 | | cncombf 25693 |
. . . . . . . 8
⊢ (((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ MblFn ∧ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))):(dom 𝐹 ∩ dom 𝐺)⟶ℂ ∧ abs ∈
(ℂ–cn→ℂ)) →
(abs ∘ (𝑧 ∈ (dom
𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn) |
| 34 | 20, 25, 32, 33 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (abs ∘ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn) |
| 35 | 24, 34 | eqeltrrd 2842 |
. . . . . 6
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn) |
| 36 | 23 | abscld 15475 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ) |
| 37 | 36 | rexrd 11311 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈
ℝ*) |
| 38 | 23 | absge0d 15483 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) |
| 39 | | elxrge0 13497 |
. . . . . . . . . . 11
⊢
((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞) ↔
((abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ ℝ* ∧ 0 ≤
(abs‘((𝐹‘𝑧) · (𝐺‘𝑧))))) |
| 40 | 37, 38, 39 | sylanbrc 583 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ∈ (0[,]+∞)) |
| 41 | | 0e0iccpnf 13499 |
. . . . . . . . . . 11
⊢ 0 ∈
(0[,]+∞) |
| 42 | 41 | a1i 11 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ∈
(0[,]+∞)) |
| 43 | 40, 42 | ifclda 4561 |
. . . . . . . . 9
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ∈
(0[,]+∞)) |
| 44 | 43 | adantr 480 |
. . . . . . . 8
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ∈
(0[,]+∞)) |
| 45 | 44 | fmpttd 7135 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))),
0)):ℝ⟶(0[,]+∞)) |
| 46 | | reex 11246 |
. . . . . . . . . . . . . . 15
⊢ ℝ
∈ V |
| 47 | 46 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → ℝ ∈
V) |
| 48 | | simprl 771 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℝ) |
| 49 | 48 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹 ∈ MblFn
∧ 𝐺 ∈
𝐿1) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → 𝑥 ∈ ℝ) |
| 50 | | elinel2 4202 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐺) |
| 51 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐺:dom 𝐺⟶ℂ ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ) |
| 52 | 7, 50, 51 | syl2an 596 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐺‘𝑧) ∈ ℂ) |
| 53 | 52 | abscld 15475 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐺‘𝑧)) ∈ ℝ) |
| 54 | 52 | absge0d 15483 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (abs‘(𝐺‘𝑧))) |
| 55 | | elrege0 13494 |
. . . . . . . . . . . . . . . . 17
⊢
((abs‘(𝐺‘𝑧)) ∈ (0[,)+∞) ↔
((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤
(abs‘(𝐺‘𝑧)))) |
| 56 | 53, 54, 55 | sylanbrc 583 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐺‘𝑧)) ∈ (0[,)+∞)) |
| 57 | | 0e0icopnf 13498 |
. . . . . . . . . . . . . . . . 17
⊢ 0 ∈
(0[,)+∞) |
| 58 | 57 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ∈
(0[,)+∞)) |
| 59 | 56, 58 | ifclda 4561 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0) ∈
(0[,)+∞)) |
| 60 | 59 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐹 ∈ MblFn
∧ 𝐺 ∈
𝐿1) ∧ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0) ∈
(0[,)+∞)) |
| 61 | | fconstmpt 5747 |
. . . . . . . . . . . . . . 15
⊢ (ℝ
× {𝑥}) = (𝑧 ∈ ℝ ↦ 𝑥) |
| 62 | 61 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (ℝ × {𝑥}) = (𝑧 ∈ ℝ ↦ 𝑥)) |
| 63 | | eqidd 2738 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) |
| 64 | 47, 49, 60, 62, 63 | offval2 7717 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((ℝ × {𝑥}) ∘f ·
(𝑧 ∈ ℝ ↦
if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) = (𝑧 ∈ ℝ ↦ (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) |
| 65 | | ovif2 7532 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)) = if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), (𝑥 · 0)) |
| 66 | 48 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝑥 ∈ ℂ) |
| 67 | 66 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℂ) |
| 68 | 67 | mul01d 11460 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · 0) = 0) |
| 69 | 68 | ifeq2d 4546 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), (𝑥 · 0)) = if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 70 | 65, 69 | eqtrid 2789 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)) = if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 71 | 70 | mpteq2dv 5244 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑧 ∈ ℝ ↦ (𝑥 · if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) |
| 72 | 64, 71 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((ℝ × {𝑥}) ∘f ·
(𝑧 ∈ ℝ ↦
if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) |
| 73 | 72 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) →
(∫2‘((ℝ × {𝑥}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) = (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))) |
| 74 | 59 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0) ∈
(0[,)+∞)) |
| 75 | 74 | fmpttd 7135 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)),
0)):ℝ⟶(0[,)+∞)) |
| 76 | 75 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)),
0)):ℝ⟶(0[,)+∞)) |
| 77 | | inss2 4238 |
. . . . . . . . . . . . . . . . . 18
⊢ (dom
𝐹 ∩ dom 𝐺) ⊆ dom 𝐺 |
| 78 | 77 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ⊆ dom 𝐺) |
| 79 | 20, 17 | mbfdm2 25672 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) |
| 80 | 7 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ dom 𝐺) → (𝐺‘𝑧) ∈ ℂ) |
| 81 | 7 | feqmptd 6977 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 = (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧))) |
| 82 | | simplr 769 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → 𝐺 ∈
𝐿1) |
| 83 | 81, 82 | eqeltrrd 2842 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ dom 𝐺 ↦ (𝐺‘𝑧)) ∈
𝐿1) |
| 84 | 78, 79, 80, 83 | iblss 25840 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (𝐺‘𝑧)) ∈
𝐿1) |
| 85 | 52, 84 | iblabs 25864 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈
𝐿1) |
| 86 | 53, 54 | iblpos 25828 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ 𝐿1 ↔
((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ MblFn ∧
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ))) |
| 87 | 85, 86 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘(𝐺‘𝑧))) ∈ MblFn ∧
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ)) |
| 88 | 87 | simprd 495 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ) |
| 89 | 88 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘(𝐺‘𝑧)), 0))) ∈ ℝ) |
| 90 | | simplrl 777 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ ℝ) |
| 91 | | neq0 4352 |
. . . . . . . . . . . . . . 15
⊢ (¬
(dom 𝐹 ∩ dom 𝐺) = ∅ ↔ ∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) |
| 92 | | 0re 11263 |
. . . . . . . . . . . . . . . . . . 19
⊢ 0 ∈
ℝ |
| 93 | 92 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ∈ ℝ) |
| 94 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 𝑧 ∈ dom 𝐹) |
| 95 | | ffvelcdm 7101 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹:dom 𝐹⟶ℂ ∧ 𝑧 ∈ dom 𝐹) → (𝐹‘𝑧) ∈ ℂ) |
| 96 | 2, 94, 95 | syl2an 596 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝐹‘𝑧) ∈ ℂ) |
| 97 | 96 | abscld 15475 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐹‘𝑧)) ∈ ℝ) |
| 98 | | simplrl 777 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 𝑥 ∈ ℝ) |
| 99 | 96 | absge0d 15483 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (abs‘(𝐹‘𝑧))) |
| 100 | | simprr 773 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑦 ∈ dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) |
| 101 | | 2fveq3 6911 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 𝑧 → (abs‘(𝐹‘𝑦)) = (abs‘(𝐹‘𝑧))) |
| 102 | 101 | breq1d 5153 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = 𝑧 → ((abs‘(𝐹‘𝑦)) ≤ 𝑥 ↔ (abs‘(𝐹‘𝑧)) ≤ 𝑥)) |
| 103 | 102 | rspccva 3621 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑦 ∈
dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 ∧ 𝑧 ∈ dom 𝐹) → (abs‘(𝐹‘𝑧)) ≤ 𝑥) |
| 104 | 100, 94, 103 | syl2an 596 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘(𝐹‘𝑧)) ≤ 𝑥) |
| 105 | 93, 97, 98, 99, 104 | letrd 11418 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ 𝑥) |
| 106 | 105 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥)) |
| 107 | 106 | exlimdv 1933 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∃𝑧 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 𝑥)) |
| 108 | 91, 107 | biimtrid 242 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ → 0 ≤ 𝑥)) |
| 109 | 108 | imp 406 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 0 ≤ 𝑥) |
| 110 | | elrege0 13494 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0[,)+∞) ↔
(𝑥 ∈ ℝ ∧ 0
≤ 𝑥)) |
| 111 | 90, 109, 110 | sylanbrc 583 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → 𝑥 ∈ (0[,)+∞)) |
| 112 | 76, 89, 111 | itg2mulc 25782 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) →
(∫2‘((ℝ × {𝑥}) ∘f · (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) = (𝑥 · (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))))) |
| 113 | 73, 112 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (𝑥 · (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0))))) |
| 114 | 90, 89 | remulcld 11291 |
. . . . . . . . . 10
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) → (𝑥 · (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘(𝐺‘𝑧)), 0)))) ∈ ℝ) |
| 115 | 113, 114 | eqeltrd 2841 |
. . . . . . . . 9
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ (dom 𝐹 ∩ dom 𝐺) = ∅) →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ) |
| 116 | 115 | ex 412 |
. . . . . . . 8
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (¬ (dom 𝐹 ∩ dom 𝐺) = ∅ →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ)) |
| 117 | | noel 4338 |
. . . . . . . . . . . . . 14
⊢ ¬
𝑧 ∈
∅ |
| 118 | | eleq2 2830 |
. . . . . . . . . . . . . 14
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↔ 𝑧 ∈ ∅)) |
| 119 | 117, 118 | mtbiri 327 |
. . . . . . . . . . . . 13
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ → ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) |
| 120 | | iffalse 4534 |
. . . . . . . . . . . . 13
⊢ (¬
𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = 0) |
| 121 | 119, 120 | syl 17 |
. . . . . . . . . . . 12
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = 0) |
| 122 | 121 | mpteq2dv 5244 |
. . . . . . . . . . 11
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ 0)) |
| 123 | | fconstmpt 5747 |
. . . . . . . . . . 11
⊢ (ℝ
× {0}) = (𝑧 ∈
ℝ ↦ 0) |
| 124 | 122, 123 | eqtr4di 2795 |
. . . . . . . . . 10
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (ℝ ×
{0})) |
| 125 | 124 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) = (∫2‘(ℝ
× {0}))) |
| 126 | | itg20 25772 |
. . . . . . . . . 10
⊢
(∫2‘(ℝ × {0})) = 0 |
| 127 | 126, 92 | eqeltri 2837 |
. . . . . . . . 9
⊢
(∫2‘(ℝ × {0})) ∈
ℝ |
| 128 | 125, 127 | eqeltrdi 2849 |
. . . . . . . 8
⊢ ((dom
𝐹 ∩ dom 𝐺) = ∅ →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ) |
| 129 | 116, 128 | pm2.61d2 181 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ) |
| 130 | 98, 53 | remulcld 11291 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ) |
| 131 | 130 | rexrd 11311 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝑥 · (abs‘(𝐺‘𝑧))) ∈
ℝ*) |
| 132 | 98, 53, 105, 54 | mulge0d 11840 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → 0 ≤ (𝑥 · (abs‘(𝐺‘𝑧)))) |
| 133 | | elxrge0 13497 |
. . . . . . . . . . . 12
⊢ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞) ↔ ((𝑥 · (abs‘(𝐺‘𝑧))) ∈ ℝ* ∧ 0 ≤
(𝑥 ·
(abs‘(𝐺‘𝑧))))) |
| 134 | 131, 132,
133 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (𝑥 · (abs‘(𝐺‘𝑧))) ∈ (0[,]+∞)) |
| 135 | 134, 42 | ifclda 4561 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) ∈
(0[,]+∞)) |
| 136 | 135 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ ℝ) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) ∈
(0[,]+∞)) |
| 137 | 136 | fmpttd 7135 |
. . . . . . . 8
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))),
0)):ℝ⟶(0[,]+∞)) |
| 138 | 96, 52 | absmuld 15493 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) = ((abs‘(𝐹‘𝑧)) · (abs‘(𝐺‘𝑧)))) |
| 139 | | abscl 15317 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝑧) ∈ ℂ → (abs‘(𝐺‘𝑧)) ∈ ℝ) |
| 140 | | absge0 15326 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝑧) ∈ ℂ → 0 ≤
(abs‘(𝐺‘𝑧))) |
| 141 | 139, 140 | jca 511 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘𝑧) ∈ ℂ → ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤
(abs‘(𝐺‘𝑧)))) |
| 142 | 52, 141 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤
(abs‘(𝐺‘𝑧)))) |
| 143 | | lemul1a 12121 |
. . . . . . . . . . . . . 14
⊢
((((abs‘(𝐹‘𝑧)) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ ((abs‘(𝐺‘𝑧)) ∈ ℝ ∧ 0 ≤
(abs‘(𝐺‘𝑧)))) ∧ (abs‘(𝐹‘𝑧)) ≤ 𝑥) → ((abs‘(𝐹‘𝑧)) · (abs‘(𝐺‘𝑧))) ≤ (𝑥 · (abs‘(𝐺‘𝑧)))) |
| 144 | 97, 98, 142, 104, 143 | syl31anc 1375 |
. . . . . . . . . . . . 13
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → ((abs‘(𝐹‘𝑧)) · (abs‘(𝐺‘𝑧))) ≤ (𝑥 · (abs‘(𝐺‘𝑧)))) |
| 145 | 138, 144 | eqbrtrd 5165 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))) ≤ (𝑥 · (abs‘(𝐺‘𝑧)))) |
| 146 | | iftrue 4531 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) |
| 147 | 146 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) |
| 148 | | iftrue 4531 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧)))) |
| 149 | 148 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0) = (𝑥 · (abs‘(𝐺‘𝑧)))) |
| 150 | 145, 147,
149 | 3brtr4d 5175 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 151 | | 0le0 12367 |
. . . . . . . . . . . . . 14
⊢ 0 ≤
0 |
| 152 | 151 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (¬
𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → 0 ≤ 0) |
| 153 | | iffalse 4534 |
. . . . . . . . . . . . 13
⊢ (¬
𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) = 0) |
| 154 | 152, 153,
120 | 3brtr4d 5175 |
. . . . . . . . . . . 12
⊢ (¬
𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 155 | 154 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) ∧ ¬ 𝑧 ∈ (dom 𝐹 ∩ dom 𝐺)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 156 | 150, 155 | pm2.61dan 813 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 157 | 156 | ralrimivw 3150 |
. . . . . . . . 9
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ∀𝑧 ∈ ℝ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) |
| 158 | 46 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ℝ ∈ V) |
| 159 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) |
| 160 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) = (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) |
| 161 | 158, 44, 136, 159, 160 | ofrfval2 7718 |
. . . . . . . . 9
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) ∘r ≤ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)) ↔ ∀𝑧 ∈ ℝ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0) ≤ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) |
| 162 | 157, 161 | mpbird 257 |
. . . . . . . 8
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) ∘r ≤ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) |
| 163 | | itg2le 25774 |
. . . . . . . 8
⊢ (((𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞) ∧
(𝑧 ∈ ℝ ↦
if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞) ∧
(𝑧 ∈ ℝ ↦
if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)) ∘r ≤ (𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ≤
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))) |
| 164 | 45, 137, 162, 163 | syl3anc 1373 |
. . . . . . 7
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ≤
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))) |
| 165 | | itg2lecl 25773 |
. . . . . . 7
⊢ (((𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0)):ℝ⟶(0[,]+∞) ∧
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0))) ∈ ℝ ∧
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ≤
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (𝑥 · (abs‘(𝐺‘𝑧))), 0)))) →
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ∈ ℝ) |
| 166 | 45, 129, 164, 165 | syl3anc 1373 |
. . . . . 6
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (∫2‘(𝑧 ∈ ℝ ↦ if(𝑧 ∈ (dom 𝐹 ∩ dom 𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ∈ ℝ) |
| 167 | 36, 38 | iblpos 25828 |
. . . . . 6
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → ((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ 𝐿1 ↔
((𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈ MblFn ∧
(∫2‘(𝑧
∈ ℝ ↦ if(𝑧
∈ (dom 𝐹 ∩ dom
𝐺), (abs‘((𝐹‘𝑧) · (𝐺‘𝑧))), 0))) ∈ ℝ))) |
| 168 | 35, 166, 167 | mpbir2and 713 |
. . . . 5
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (abs‘((𝐹‘𝑧) · (𝐺‘𝑧)))) ∈
𝐿1) |
| 169 | 17, 20, 168 | iblabsr 25865 |
. . . 4
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝑧 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑧) · (𝐺‘𝑧))) ∈
𝐿1) |
| 170 | 16, 169 | eqeltrd 2841 |
. . 3
⊢ (((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
∧ (𝑥 ∈ ℝ
∧ ∀𝑦 ∈ dom
𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥)) → (𝐹 ∘f · 𝐺) ∈
𝐿1) |
| 171 | 170 | rexlimdvaa 3156 |
. 2
⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1)
→ (∃𝑥 ∈
ℝ ∀𝑦 ∈
dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥 → (𝐹 ∘f · 𝐺) ∈
𝐿1)) |
| 172 | 171 | 3impia 1118 |
1
⊢ ((𝐹 ∈ MblFn ∧ 𝐺 ∈ 𝐿1
∧ ∃𝑥 ∈
ℝ ∀𝑦 ∈
dom 𝐹(abs‘(𝐹‘𝑦)) ≤ 𝑥) → (𝐹 ∘f · 𝐺) ∈
𝐿1) |