| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | mblss 25567 | . . . 4
⊢ (𝐴 ∈ dom vol → 𝐴 ⊆
ℝ) | 
| 2 |  | mblss 25567 | . . . 4
⊢ (𝐵 ∈ dom vol → 𝐵 ⊆
ℝ) | 
| 3 | 1, 2 | anim12i 613 | . . 3
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ⊆ ℝ ∧ 𝐵 ⊆
ℝ)) | 
| 4 |  | unss 4189 | . . 3
⊢ ((𝐴 ⊆ ℝ ∧ 𝐵 ⊆ ℝ) ↔ (𝐴 ∪ 𝐵) ⊆ ℝ) | 
| 5 | 3, 4 | sylib 218 | . 2
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ⊆ ℝ) | 
| 6 |  | elpwi 4606 | . . . 4
⊢ (𝑥 ∈ 𝒫 ℝ →
𝑥 ⊆
ℝ) | 
| 7 |  | inss1 4236 | . . . . . . . . 9
⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥 | 
| 8 |  | ovolsscl 25522 | . . . . . . . . 9
⊢ (((𝑥 ∩ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ) | 
| 9 | 7, 8 | mp3an1 1449 | . . . . . . . 8
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ) | 
| 10 | 9 | adantl 481 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ∈ ℝ) | 
| 11 |  | inss1 4236 | . . . . . . . . . 10
⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥 | 
| 12 |  | ovolsscl 25522 | . . . . . . . . . 10
⊢ (((𝑥 ∩ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ 𝐴)) ∈
ℝ) | 
| 13 | 11, 12 | mp3an1 1449 | . . . . . . . . 9
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ) | 
| 14 | 13 | adantl 481 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ) | 
| 15 |  | inss1 4236 | . . . . . . . . 9
⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴) | 
| 16 |  | difss 4135 | . . . . . . . . . 10
⊢ (𝑥 ∖ 𝐴) ⊆ 𝑥 | 
| 17 |  | simprl 770 | . . . . . . . . . 10
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → 𝑥 ⊆
ℝ) | 
| 18 | 16, 17 | sstrid 3994 | . . . . . . . . 9
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (𝑥 ∖
𝐴) ⊆
ℝ) | 
| 19 |  | ovolsscl 25522 | . . . . . . . . . . 11
⊢ (((𝑥 ∖ 𝐴) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∖
𝐴)) ∈
ℝ) | 
| 20 | 16, 19 | mp3an1 1449 | . . . . . . . . . 10
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) | 
| 21 | 20 | adantl 481 | . . . . . . . . 9
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) | 
| 22 |  | ovolsscl 25522 | . . . . . . . . 9
⊢ ((((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∖ 𝐴) ∧ (𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) →
(vol*‘((𝑥 ∖
𝐴) ∩ 𝐵)) ∈ ℝ) | 
| 23 | 15, 18, 21, 22 | mp3an2i 1467 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ) | 
| 24 | 14, 23 | readdcld 11291 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) ∈ ℝ) | 
| 25 |  | difss 4135 | . . . . . . . . 9
⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥 | 
| 26 |  | ovolsscl 25522 | . . . . . . . . 9
⊢ (((𝑥 ∖ (𝐴 ∪ 𝐵)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∖
(𝐴 ∪ 𝐵))) ∈ ℝ) | 
| 27 | 25, 26 | mp3an1 1449 | . . . . . . . 8
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ) | 
| 28 | 27 | adantl 481 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℝ) | 
| 29 |  | incom 4208 | . . . . . . . . . . . 12
⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝐵 ∩ (𝑥 ∖ 𝐴)) | 
| 30 |  | indifcom 4282 | . . . . . . . . . . . 12
⊢ (𝐵 ∩ (𝑥 ∖ 𝐴)) = (𝑥 ∩ (𝐵 ∖ 𝐴)) | 
| 31 | 29, 30 | eqtri 2764 | . . . . . . . . . . 11
⊢ ((𝑥 ∖ 𝐴) ∩ 𝐵) = (𝑥 ∩ (𝐵 ∖ 𝐴)) | 
| 32 | 31 | uneq2i 4164 | . . . . . . . . . 10
⊢ ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴))) | 
| 33 |  | indi 4283 | . . . . . . . . . 10
⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = ((𝑥 ∩ 𝐴) ∪ (𝑥 ∩ (𝐵 ∖ 𝐴))) | 
| 34 |  | undif2 4476 | . . . . . . . . . . 11
⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵) | 
| 35 | 34 | ineq2i 4216 | . . . . . . . . . 10
⊢ (𝑥 ∩ (𝐴 ∪ (𝐵 ∖ 𝐴))) = (𝑥 ∩ (𝐴 ∪ 𝐵)) | 
| 36 | 32, 33, 35 | 3eqtr2ri 2771 | . . . . . . . . 9
⊢ (𝑥 ∩ (𝐴 ∪ 𝐵)) = ((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵)) | 
| 37 | 36 | fveq2i 6908 | . . . . . . . 8
⊢
(vol*‘(𝑥 ∩
(𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))) | 
| 38 | 11, 17 | sstrid 3994 | . . . . . . . . 9
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (𝑥 ∩
𝐴) ⊆
ℝ) | 
| 39 | 15, 18 | sstrid 3994 | . . . . . . . . 9
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → ((𝑥
∖ 𝐴) ∩ 𝐵) ⊆
ℝ) | 
| 40 |  | ovolun 25535 | . . . . . . . . 9
⊢ ((((𝑥 ∩ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∩ 𝐴)) ∈ ℝ) ∧ (((𝑥 ∖ 𝐴) ∩ 𝐵) ⊆ ℝ ∧ (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℝ)) →
(vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)))) | 
| 41 | 38, 14, 39, 23, 40 | syl22anc 838 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘((𝑥 ∩ 𝐴) ∪ ((𝑥 ∖ 𝐴) ∩ 𝐵))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)))) | 
| 42 | 37, 41 | eqbrtrid 5177 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) ≤ ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)))) | 
| 43 | 10, 24, 28, 42 | leadd1dd 11878 | . . . . . 6
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))))) | 
| 44 |  | simplr 768 | . . . . . . . . . 10
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → 𝐵 ∈
dom vol) | 
| 45 |  | mblsplit 25568 | . . . . . . . . . 10
⊢ ((𝐵 ∈ dom vol ∧ (𝑥 ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ 𝐴)) ∈ ℝ) → (vol*‘(𝑥 ∖ 𝐴)) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)))) | 
| 46 | 44, 18, 21, 45 | syl3anc 1372 | . . . . . . . . 9
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)))) | 
| 47 |  | difun1 4298 | . . . . . . . . . . 11
⊢ (𝑥 ∖ (𝐴 ∪ 𝐵)) = ((𝑥 ∖ 𝐴) ∖ 𝐵) | 
| 48 | 47 | fveq2i 6908 | . . . . . . . . . 10
⊢
(vol*‘(𝑥
∖ (𝐴 ∪ 𝐵))) = (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵)) | 
| 49 | 48 | oveq2i 7443 | . . . . . . . . 9
⊢
((vol*‘((𝑥
∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘((𝑥 ∖ 𝐴) ∖ 𝐵))) | 
| 50 | 46, 49 | eqtr4di 2794 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∖ 𝐴)) = ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))))) | 
| 51 | 50 | oveq2d 7448 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴))) = ((vol*‘(𝑥 ∩ 𝐴)) + ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))))) | 
| 52 |  | simpll 766 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → 𝐴 ∈
dom vol) | 
| 53 |  | simprr 772 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘𝑥) ∈ ℝ) | 
| 54 |  | mblsplit 25568 | . . . . . . . 8
⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) | 
| 55 | 52, 17, 53, 54 | syl3anc 1372 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘(𝑥 ∖ 𝐴)))) | 
| 56 | 14 | recnd 11290 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∩ 𝐴)) ∈ ℂ) | 
| 57 | 23 | recnd 11290 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) ∈ ℂ) | 
| 58 | 28 | recnd 11290 | . . . . . . . 8
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))) ∈ ℂ) | 
| 59 | 56, 57, 58 | addassd 11284 | . . . . . . 7
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) = ((vol*‘(𝑥 ∩ 𝐴)) + ((vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵)) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))))) | 
| 60 | 51, 55, 59 | 3eqtr4d 2786 | . . . . . 6
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → (vol*‘𝑥) = (((vol*‘(𝑥 ∩ 𝐴)) + (vol*‘((𝑥 ∖ 𝐴) ∩ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵))))) | 
| 61 | 43, 60 | breqtrrd 5170 | . . . . 5
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ (𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ)) → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥)) | 
| 62 | 61 | expr 456 | . . . 4
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ⊆ ℝ) →
((vol*‘𝑥) ∈
ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥))) | 
| 63 | 6, 62 | sylan2 593 | . . 3
⊢ (((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) ∧ 𝑥 ∈ 𝒫 ℝ)
→ ((vol*‘𝑥)
∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥))) | 
| 64 | 63 | ralrimiva 3145 | . 2
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) →
∀𝑥 ∈ 𝒫
ℝ((vol*‘𝑥)
∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥))) | 
| 65 |  | ismbl2 25563 | . 2
⊢ ((𝐴 ∪ 𝐵) ∈ dom vol ↔ ((𝐴 ∪ 𝐵) ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫
ℝ((vol*‘𝑥)
∈ ℝ → ((vol*‘(𝑥 ∩ (𝐴 ∪ 𝐵))) + (vol*‘(𝑥 ∖ (𝐴 ∪ 𝐵)))) ≤ (vol*‘𝑥)))) | 
| 66 | 5, 64, 65 | sylanbrc 583 | 1
⊢ ((𝐴 ∈ dom vol ∧ 𝐵 ∈ dom vol) → (𝐴 ∪ 𝐵) ∈ dom vol) |