| Step | Hyp | Ref
| Expression |
| 1 | | mbfposadd.2 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
| 2 | | 0re 11263 |
. . . . 5
⊢ 0 ∈
ℝ |
| 3 | | ifcl 4571 |
. . . . 5
⊢ ((𝐵 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ) |
| 4 | 1, 2, 3 | sylancl 586 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℝ) |
| 5 | | mbfposadd.4 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℝ) |
| 6 | | ifcl 4571 |
. . . . 5
⊢ ((𝐶 ∈ ℝ ∧ 0 ∈
ℝ) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) |
| 7 | 5, 2, 6 | sylancl 586 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐶, 𝐶, 0) ∈ ℝ) |
| 8 | 4, 7 | readdcld 11290 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ∈ ℝ) |
| 9 | 8 | fmpttd 7135 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ) |
| 10 | | ssrab2 4080 |
. . . 4
⊢ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴 |
| 11 | | fssres 6774 |
. . . 4
⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))):𝐴⟶ℝ ∧ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ⊆ 𝐴) → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}):{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}⟶ℝ) |
| 12 | 9, 10, 11 | sylancl 586 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}):{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}⟶ℝ) |
| 13 | | inss2 4238 |
. . . . . 6
⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} |
| 14 | | resabs1 6024 |
. . . . . 6
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))) |
| 15 | 13, 14 | ax-mp 5 |
. . . . 5
⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 16 | | elin 3967 |
. . . . . . . . 9
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 17 | | rabid 3458 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵)) |
| 18 | | rabid 3458 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) |
| 19 | 17, 18 | anbi12i 628 |
. . . . . . . . 9
⊢ ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))) |
| 20 | 16, 19 | bitri 275 |
. . . . . . . 8
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))) |
| 21 | | iftrue 4531 |
. . . . . . . . . 10
⊢ (0 ≤
𝐵 → if(0 ≤ 𝐵, 𝐵, 0) = 𝐵) |
| 22 | | iftrue 4531 |
. . . . . . . . . 10
⊢ (0 ≤
𝐶 → if(0 ≤ 𝐶, 𝐶, 0) = 𝐶) |
| 23 | 21, 22 | oveqan12d 7450 |
. . . . . . . . 9
⊢ ((0 ≤
𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶)) |
| 24 | 23 | ad2ant2l 746 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶)) |
| 25 | 20, 24 | sylbi 217 |
. . . . . . 7
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (𝐵 + 𝐶)) |
| 26 | 25 | mpteq2ia 5245 |
. . . . . 6
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) |
| 27 | | inss1 4237 |
. . . . . . . 8
⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} |
| 28 | | ssrab2 4080 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ⊆ 𝐴 |
| 29 | 27, 28 | sstri 3993 |
. . . . . . 7
⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 |
| 30 | | resmpt 6055 |
. . . . . . . 8
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 31 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(if(0
≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) |
| 32 | | nfcsb1v 3923 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) |
| 33 | | csbeq1a 3913 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 34 | 31, 32, 33 | cbvmpt 5253 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 35 | 34 | reseq1i 5993 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 36 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 37 | | nfrab1 3457 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} |
| 38 | | nfrab1 3457 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} |
| 39 | 37, 38 | nfin 4224 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 40 | 39 | nfcri 2897 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 41 | 32 | nfeq2 2923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) |
| 42 | 40, 41 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 43 | | eleq1w 2824 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))) |
| 44 | 33 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) ↔ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 45 | 43, 44 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))) |
| 46 | 36, 42, 45 | cbvopab1 5217 |
. . . . . . . . 9
⊢
{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 47 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 48 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 49 | 46, 47, 48 | 3eqtr4i 2775 |
. . . . . . . 8
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 50 | 30, 35, 49 | 3eqtr4g 2802 |
. . . . . . 7
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 51 | 29, 50 | ax-mp 5 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 52 | | resmpt 6055 |
. . . . . . . 8
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))) |
| 53 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝐵 + 𝐶) |
| 54 | | nfcsb1v 3923 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌(𝐵 + 𝐶) |
| 55 | | csbeq1a 3913 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝐵 + 𝐶) = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) |
| 56 | 53, 54, 55 | cbvmpt 5253 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) |
| 57 | 56 | reseq1i 5993 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 58 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶)) |
| 59 | 54 | nfeq2 2923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶) |
| 60 | 40, 59 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) |
| 61 | 55 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑧 = (𝐵 + 𝐶) ↔ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))) |
| 62 | 43, 61 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶)) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)))) |
| 63 | 58, 60, 62 | cbvopab1 5217 |
. . . . . . . . 9
⊢
{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))} |
| 64 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (𝐵 + 𝐶))} |
| 65 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶))} |
| 66 | 63, 64, 65 | 3eqtr4i 2775 |
. . . . . . . 8
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(𝐵 + 𝐶)) |
| 67 | 52, 57, 66 | 3eqtr4g 2802 |
. . . . . . 7
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶))) |
| 68 | 29, 67 | ax-mp 5 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (𝐵 + 𝐶)) |
| 69 | 26, 51, 68 | 3eqtr4i 2775 |
. . . . 5
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 70 | 15, 69 | eqtri 2765 |
. . . 4
⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 71 | | mbfposadd.5 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn) |
| 72 | 1 | biantrurd 532 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵))) |
| 73 | | elrege0 13494 |
. . . . . . . . . 10
⊢ (𝐵 ∈ (0[,)+∞) ↔
(𝐵 ∈ ℝ ∧ 0
≤ 𝐵)) |
| 74 | 72, 73 | bitr4di 289 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐵 ↔ 𝐵 ∈ (0[,)+∞))) |
| 75 | 74 | rabbidva 3443 |
. . . . . . . 8
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (0[,)+∞)}) |
| 76 | | 0xr 11308 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ* |
| 77 | | pnfxr 11315 |
. . . . . . . . . . 11
⊢ +∞
∈ ℝ* |
| 78 | | 0ltpnf 13164 |
. . . . . . . . . . 11
⊢ 0 <
+∞ |
| 79 | | snunioo 13518 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0
< +∞) → ({0} ∪ (0(,)+∞)) =
(0[,)+∞)) |
| 80 | 76, 77, 78, 79 | mp3an 1463 |
. . . . . . . . . 10
⊢ ({0}
∪ (0(,)+∞)) = (0[,)+∞) |
| 81 | 80 | imaeq2i 6076 |
. . . . . . . . 9
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ ({0} ∪ (0(,)+∞))) =
(◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0[,)+∞)) |
| 82 | | imaundi 6169 |
. . . . . . . . 9
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ ({0} ∪ (0(,)+∞))) =
((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) |
| 83 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| 84 | 83 | mptpreima 6258 |
. . . . . . . . 9
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0[,)+∞)) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (0[,)+∞)} |
| 85 | 81, 82, 84 | 3eqtr3ri 2774 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (0[,)+∞)} = ((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) |
| 86 | 75, 85 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} = ((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)))) |
| 87 | | mbfposadd.1 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
| 88 | 1 | fmpttd 7135 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
| 89 | | mbfimasn 25667 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ ∧ 0 ∈ ℝ)
→ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∈ dom
vol) |
| 90 | 2, 89 | mp3an3 1452 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∈ dom
vol) |
| 91 | | mbfima 25665 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)) ∈ dom
vol) |
| 92 | | unmbl 25572 |
. . . . . . . . 9
⊢ (((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∈ dom vol ∧ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞)) ∈ dom vol)
→ ((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) ∈ dom
vol) |
| 93 | 90, 91, 92 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → ((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) ∈ dom
vol) |
| 94 | 87, 88, 93 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (0(,)+∞))) ∈ dom
vol) |
| 95 | 86, 94 | eqeltrd 2841 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol) |
| 96 | 5 | biantrurd 532 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐶 ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))) |
| 97 | | elrege0 13494 |
. . . . . . . . . 10
⊢ (𝐶 ∈ (0[,)+∞) ↔
(𝐶 ∈ ℝ ∧ 0
≤ 𝐶)) |
| 98 | 96, 97 | bitr4di 289 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 ≤ 𝐶 ↔ 𝐶 ∈ (0[,)+∞))) |
| 99 | 98 | rabbidva 3443 |
. . . . . . . 8
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (0[,)+∞)}) |
| 100 | 80 | imaeq2i 6076 |
. . . . . . . . 9
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ ({0} ∪ (0(,)+∞))) =
(◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0[,)+∞)) |
| 101 | | imaundi 6169 |
. . . . . . . . 9
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ ({0} ∪ (0(,)+∞))) =
((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) |
| 102 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
| 103 | 102 | mptpreima 6258 |
. . . . . . . . 9
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0[,)+∞)) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (0[,)+∞)} |
| 104 | 100, 101,
103 | 3eqtr3ri 2774 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (0[,)+∞)} = ((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) |
| 105 | 99, 104 | eqtrdi 2793 |
. . . . . . 7
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = ((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)))) |
| 106 | | mbfposadd.3 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) |
| 107 | 5 | fmpttd 7135 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) |
| 108 | | mbfimasn 25667 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ ∧ 0 ∈ ℝ)
→ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∈ dom
vol) |
| 109 | 2, 108 | mp3an3 1452 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∈ dom
vol) |
| 110 | | mbfima 25665 |
. . . . . . . . 9
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)) ∈ dom
vol) |
| 111 | | unmbl 25572 |
. . . . . . . . 9
⊢ (((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∈ dom vol ∧ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞)) ∈ dom vol)
→ ((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) ∈ dom
vol) |
| 112 | 109, 110,
111 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → ((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) ∈ dom
vol) |
| 113 | 106, 107,
112 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ {0}) ∪ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (0(,)+∞))) ∈ dom
vol) |
| 114 | 105, 113 | eqeltrd 2841 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) |
| 115 | | inmbl 25577 |
. . . . . 6
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) |
| 116 | 95, 114, 115 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) |
| 117 | | mbfres 25679 |
. . . . 5
⊢ (((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ∈ MblFn ∧ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn) |
| 118 | 71, 116, 117 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐵 + 𝐶)) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn) |
| 119 | 70, 118 | eqeltrid 2845 |
. . 3
⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn) |
| 120 | | inss2 4238 |
. . . . . 6
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} |
| 121 | | resabs1 6024 |
. . . . . 6
⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))) |
| 122 | 120, 121 | ax-mp 5 |
. . . . 5
⊢ (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 123 | | rabid 3458 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐵)) |
| 124 | 123, 18 | anbi12i 628 |
. . . . . . . . 9
⊢ ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))) |
| 125 | | elin 3967 |
. . . . . . . . 9
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 126 | | anandi 676 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝐶))) |
| 127 | 124, 125,
126 | 3bitr4i 303 |
. . . . . . . 8
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) |
| 128 | | iffalse 4534 |
. . . . . . . . . . 11
⊢ (¬ 0
≤ 𝐵 → if(0 ≤
𝐵, 𝐵, 0) = 0) |
| 129 | 128, 22 | oveqan12d 7450 |
. . . . . . . . . 10
⊢ ((¬ 0
≤ 𝐵 ∧ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶)) |
| 130 | 129 | ad2antll 729 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (0 + 𝐶)) |
| 131 | 5 | recnd 11289 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 132 | 131 | addlidd 11462 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (0 + 𝐶) = 𝐶) |
| 133 | 132 | adantrr 717 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (0 + 𝐶) = 𝐶) |
| 134 | 130, 133 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ (¬ 0 ≤ 𝐵 ∧ 0 ≤ 𝐶))) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶) |
| 135 | 127, 134 | sylan2b 594 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = 𝐶) |
| 136 | 135 | mpteq2dva 5242 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶)) |
| 137 | | inss1 4237 |
. . . . . . . 8
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} |
| 138 | | ssrab2 4080 |
. . . . . . . 8
⊢ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ⊆ 𝐴 |
| 139 | 137, 138 | sstri 3993 |
. . . . . . 7
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 |
| 140 | | resmpt 6055 |
. . . . . . . 8
⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 141 | 34 | reseq1i 5993 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 142 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 143 | | nfrab1 3457 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} |
| 144 | 143, 38 | nfin 4224 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 145 | 144 | nfcri 2897 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 146 | 145, 41 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 147 | | eleq1w 2824 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↔ 𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))) |
| 148 | 147, 44 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))) |
| 149 | 142, 146,
148 | cbvopab1 5217 |
. . . . . . . . 9
⊢
{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 150 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 151 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 152 | 149, 150,
151 | 3eqtr4i 2775 |
. . . . . . . 8
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 153 | 140, 141,
152 | 3eqtr4g 2802 |
. . . . . . 7
⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 154 | 139, 153 | ax-mp 5 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 155 | | resmpt 6055 |
. . . . . . . 8
⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌𝐶)) |
| 156 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑦𝐶 |
| 157 | | nfcsb1v 3923 |
. . . . . . . . . 10
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 |
| 158 | | csbeq1a 3913 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶) |
| 159 | 156, 157,
158 | cbvmpt 5253 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) |
| 160 | 159 | reseq1i 5993 |
. . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 161 | | nfv 1914 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶) |
| 162 | 157 | nfeq2 2923 |
. . . . . . . . . . 11
⊢
Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌𝐶 |
| 163 | 145, 162 | nfan 1899 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶) |
| 164 | 158 | eqeq2d 2748 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)) |
| 165 | 147, 164 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶) ↔ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶))) |
| 166 | 161, 163,
165 | cbvopab1 5217 |
. . . . . . . . 9
⊢
{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)} |
| 167 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = 𝐶)} |
| 168 | | df-mpt 5226 |
. . . . . . . . 9
⊢ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌𝐶) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∧ 𝑧 = ⦋𝑦 / 𝑥⦌𝐶)} |
| 169 | 166, 167,
168 | 3eqtr4i 2775 |
. . . . . . . 8
⊢ (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) = (𝑦 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ ⦋𝑦 / 𝑥⦌𝐶) |
| 170 | 155, 160,
169 | 3eqtr4g 2802 |
. . . . . . 7
⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶)) |
| 171 | 139, 170 | ax-mp 5 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = (𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↦ 𝐶) |
| 172 | 136, 154,
171 | 3eqtr4g 2802 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))) |
| 173 | 122, 172 | eqtrid 2789 |
. . . 4
⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}))) |
| 174 | 83 | mptpreima 6258 |
. . . . . . . 8
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (-∞(,)0)} |
| 175 | | elioomnf 13484 |
. . . . . . . . . . 11
⊢ (0 ∈
ℝ* → (𝐵 ∈ (-∞(,)0) ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0))) |
| 176 | 76, 175 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝐵 ∈ (-∞(,)0) ↔
(𝐵 ∈ ℝ ∧
𝐵 < 0)) |
| 177 | 1 | biantrurd 532 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 < 0 ↔ (𝐵 ∈ ℝ ∧ 𝐵 < 0))) |
| 178 | | ltnle 11340 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℝ ∧ 0 ∈
ℝ) → (𝐵 < 0
↔ ¬ 0 ≤ 𝐵)) |
| 179 | 1, 2, 178 | sylancl 586 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 < 0 ↔ ¬ 0 ≤ 𝐵)) |
| 180 | 177, 179 | bitr3d 281 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐵 ∈ ℝ ∧ 𝐵 < 0) ↔ ¬ 0 ≤ 𝐵)) |
| 181 | 176, 180 | bitrid 283 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 ∈ (-∞(,)0) ↔ ¬ 0 ≤
𝐵)) |
| 182 | 181 | rabbidva 3443 |
. . . . . . . 8
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (-∞(,)0)} = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) |
| 183 | 174, 182 | eqtrid 2789 |
. . . . . . 7
⊢ (𝜑 → (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) |
| 184 | | mbfima 25665 |
. . . . . . . 8
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) → (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) ∈ dom
vol) |
| 185 | 87, 88, 184 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (◡(𝑥 ∈ 𝐴 ↦ 𝐵) “ (-∞(,)0)) ∈ dom
vol) |
| 186 | 183, 185 | eqeltrrd 2842 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol) |
| 187 | | inmbl 25577 |
. . . . . 6
⊢ (({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∈ dom vol ∧ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∈ dom vol) → ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) |
| 188 | 186, 114,
187 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) |
| 189 | | mbfres 25679 |
. . . . 5
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ dom vol) → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn) |
| 190 | 106, 188,
189 | syl2anc 584 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn) |
| 191 | 173, 190 | eqeltrd 2841 |
. . 3
⊢ (𝜑 → (((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ↾ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) ∈ MblFn) |
| 192 | | ssid 4006 |
. . . . . 6
⊢ 𝐴 ⊆ 𝐴 |
| 193 | | dfrab3ss 4323 |
. . . . . 6
⊢ (𝐴 ⊆ 𝐴 → {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 194 | 192, 193 | ax-mp 5 |
. . . . 5
⊢ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} = (𝐴 ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 195 | | rabxm 4390 |
. . . . . 6
⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) |
| 196 | 195 | ineq1i 4216 |
. . . . 5
⊢ (𝐴 ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 197 | | indir 4286 |
. . . . 5
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵}) ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) = (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) |
| 198 | 194, 196,
197 | 3eqtrri 2770 |
. . . 4
⊢ (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} |
| 199 | 198 | a1i 11 |
. . 3
⊢ (𝜑 → (({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∪ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐵} ∩ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶})) = {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) |
| 200 | 12, 119, 191, 199 | mbfres2 25680 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶}) ∈ MblFn) |
| 201 | | rabid 3458 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐶)) |
| 202 | | iffalse 4534 |
. . . . . . . . 9
⊢ (¬ 0
≤ 𝐶 → if(0 ≤
𝐶, 𝐶, 0) = 0) |
| 203 | 202 | oveq2d 7447 |
. . . . . . . 8
⊢ (¬ 0
≤ 𝐶 → (if(0 ≤
𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = (if(0 ≤ 𝐵, 𝐵, 0) + 0)) |
| 204 | 4 | recnd 11289 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → if(0 ≤ 𝐵, 𝐵, 0) ∈ ℂ) |
| 205 | 204 | addridd 11461 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (if(0 ≤ 𝐵, 𝐵, 0) + 0) = if(0 ≤ 𝐵, 𝐵, 0)) |
| 206 | 203, 205 | sylan9eqr 2799 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ ¬ 0 ≤ 𝐶) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0)) |
| 207 | 206 | anasss 466 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ ¬ 0 ≤ 𝐶)) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0)) |
| 208 | 201, 207 | sylan2b 594 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) → (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)) = if(0 ≤ 𝐵, 𝐵, 0)) |
| 209 | 208 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
| 210 | | ssrab2 4080 |
. . . . 5
⊢ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 |
| 211 | | resmpt 6055 |
. . . . . 6
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 212 | 34 | reseq1i 5993 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) |
| 213 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 214 | | nfrab1 3457 |
. . . . . . . . . 10
⊢
Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} |
| 215 | 214 | nfcri 2897 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} |
| 216 | 215, 41 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 217 | | eleq1w 2824 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})) |
| 218 | 217, 44 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↔ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))))) |
| 219 | 213, 216,
218 | cbvopab1 5217 |
. . . . . . 7
⊢
{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 220 | | df-mpt 5226 |
. . . . . . 7
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 221 | | df-mpt 5226 |
. . . . . . 7
⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))} |
| 222 | 219, 220,
221 | 3eqtr4i 2775 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌(if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 223 | 211, 212,
222 | 3eqtr4g 2802 |
. . . . 5
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0)))) |
| 224 | 210, 223 | ax-mp 5 |
. . . 4
⊢ ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) |
| 225 | | resmpt 6055 |
. . . . . 6
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))) |
| 226 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑦if(0
≤ 𝐵, 𝐵, 0) |
| 227 | | nfcsb1v 3923 |
. . . . . . . 8
⊢
Ⅎ𝑥⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0) |
| 228 | | csbeq1a 3913 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → if(0 ≤ 𝐵, 𝐵, 0) = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) |
| 229 | 226, 227,
228 | cbvmpt 5253 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) |
| 230 | 229 | reseq1i 5993 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) |
| 231 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0)) |
| 232 | 227 | nfeq2 2923 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0) |
| 233 | 215, 232 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) |
| 234 | 228 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑧 = if(0 ≤ 𝐵, 𝐵, 0) ↔ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))) |
| 235 | 217, 234 | anbi12d 632 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0)) ↔ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)))) |
| 236 | 231, 233,
235 | cbvopab1 5217 |
. . . . . . 7
⊢
{〈𝑥, 𝑧〉 ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))} = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))} |
| 237 | | df-mpt 5226 |
. . . . . . 7
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = {〈𝑥, 𝑧〉 ∣ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = if(0 ≤ 𝐵, 𝐵, 0))} |
| 238 | | df-mpt 5226 |
. . . . . . 7
⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) = {〈𝑦, 𝑧〉 ∣ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∧ 𝑧 = ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0))} |
| 239 | 236, 237,
238 | 3eqtr4i 2775 |
. . . . . 6
⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) = (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ ⦋𝑦 / 𝑥⦌if(0 ≤ 𝐵, 𝐵, 0)) |
| 240 | 225, 230,
239 | 3eqtr4g 2802 |
. . . . 5
⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0))) |
| 241 | 210, 240 | ax-mp 5 |
. . . 4
⊢ ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ↦ if(0 ≤ 𝐵, 𝐵, 0)) |
| 242 | 209, 224,
241 | 3eqtr4g 2802 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶})) |
| 243 | 1, 87 | mbfpos 25686 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn) |
| 244 | 102 | mptpreima 6258 |
. . . . . 6
⊢ (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (-∞(,)0)} |
| 245 | | elioomnf 13484 |
. . . . . . . . 9
⊢ (0 ∈
ℝ* → (𝐶 ∈ (-∞(,)0) ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0))) |
| 246 | 76, 245 | ax-mp 5 |
. . . . . . . 8
⊢ (𝐶 ∈ (-∞(,)0) ↔
(𝐶 ∈ ℝ ∧
𝐶 < 0)) |
| 247 | 5 | biantrurd 532 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 < 0 ↔ (𝐶 ∈ ℝ ∧ 𝐶 < 0))) |
| 248 | | ltnle 11340 |
. . . . . . . . . 10
⊢ ((𝐶 ∈ ℝ ∧ 0 ∈
ℝ) → (𝐶 < 0
↔ ¬ 0 ≤ 𝐶)) |
| 249 | 5, 2, 248 | sylancl 586 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 < 0 ↔ ¬ 0 ≤ 𝐶)) |
| 250 | 247, 249 | bitr3d 281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐶 ∈ ℝ ∧ 𝐶 < 0) ↔ ¬ 0 ≤ 𝐶)) |
| 251 | 246, 250 | bitrid 283 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ∈ (-∞(,)0) ↔ ¬ 0 ≤
𝐶)) |
| 252 | 251 | rabbidva 3443 |
. . . . . 6
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ∈ (-∞(,)0)} = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) |
| 253 | 244, 252 | eqtrid 2789 |
. . . . 5
⊢ (𝜑 → (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) = {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) |
| 254 | | mbfima 25665 |
. . . . . 6
⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶ℝ) → (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) ∈ dom
vol) |
| 255 | 106, 107,
254 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (◡(𝑥 ∈ 𝐴 ↦ 𝐶) “ (-∞(,)0)) ∈ dom
vol) |
| 256 | 253, 255 | eqeltrrd 2842 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol) |
| 257 | | mbfres 25679 |
. . . 4
⊢ (((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ∈ MblFn ∧ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶} ∈ dom vol) → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn) |
| 258 | 243, 256,
257 | syl2anc 584 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ if(0 ≤ 𝐵, 𝐵, 0)) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn) |
| 259 | 242, 258 | eqeltrd 2841 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ↾ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) ∈ MblFn) |
| 260 | | rabxm 4390 |
. . . 4
⊢ 𝐴 = ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) |
| 261 | 260 | eqcomi 2746 |
. . 3
⊢ ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴 |
| 262 | 261 | a1i 11 |
. 2
⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 0 ≤ 𝐶} ∪ {𝑥 ∈ 𝐴 ∣ ¬ 0 ≤ 𝐶}) = 𝐴) |
| 263 | 9, 200, 259, 262 | mbfres2 25680 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (if(0 ≤ 𝐵, 𝐵, 0) + if(0 ≤ 𝐶, 𝐶, 0))) ∈ MblFn) |