| Step | Hyp | Ref
| Expression |
| 1 | | voliunlem.3 |
. . . . 5
⊢ (𝜑 → 𝐹:ℕ⟶dom vol) |
| 2 | 1 | frnd 6744 |
. . . 4
⊢ (𝜑 → ran 𝐹 ⊆ dom vol) |
| 3 | | mblss 25566 |
. . . . . 6
⊢ (𝑥 ∈ dom vol → 𝑥 ⊆
ℝ) |
| 4 | | velpw 4605 |
. . . . . 6
⊢ (𝑥 ∈ 𝒫 ℝ ↔
𝑥 ⊆
ℝ) |
| 5 | 3, 4 | sylibr 234 |
. . . . 5
⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫
ℝ) |
| 6 | 5 | ssriv 3987 |
. . . 4
⊢ dom vol
⊆ 𝒫 ℝ |
| 7 | 2, 6 | sstrdi 3996 |
. . 3
⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ) |
| 8 | | sspwuni 5100 |
. . 3
⊢ (ran
𝐹 ⊆ 𝒫 ℝ
↔ ∪ ran 𝐹 ⊆ ℝ) |
| 9 | 7, 8 | sylib 218 |
. 2
⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ) |
| 10 | | elpwi 4607 |
. . . 4
⊢ (𝑥 ∈ 𝒫 ℝ →
𝑥 ⊆
ℝ) |
| 11 | | inundif 4479 |
. . . . . . . 8
⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹)) = 𝑥 |
| 12 | 11 | fveq2i 6909 |
. . . . . . 7
⊢
(vol*‘((𝑥
∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹))) = (vol*‘𝑥) |
| 13 | | inss1 4237 |
. . . . . . . . 9
⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 |
| 14 | | simp2 1138 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆
ℝ) |
| 15 | 13, 14 | sstrid 3995 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ) |
| 16 | | ovolsscl 25521 |
. . . . . . . . . 10
⊢ (((𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) |
| 17 | 13, 16 | mp3an1 1450 |
. . . . . . . . 9
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈
ℝ) |
| 18 | 17 | 3adant1 1131 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) |
| 19 | | difss 4136 |
. . . . . . . . 9
⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 |
| 20 | 19, 14 | sstrid 3995 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ) |
| 21 | | ovolsscl 25521 |
. . . . . . . . . 10
⊢ (((𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ) |
| 22 | 19, 21 | mp3an1 1450 |
. . . . . . . . 9
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∖ ∪ ran
𝐹)) ∈
ℝ) |
| 23 | 22 | 3adant1 1131 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ) |
| 24 | | ovolun 25534 |
. . . . . . . 8
⊢ ((((𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ ∧ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) ∧ ((𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)) →
(vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹))) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 25 | 15, 18, 20, 23, 24 | syl22anc 839 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹))) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 26 | 12, 25 | eqbrtrrid 5179 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘𝑥) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 27 | 18 | rexrd 11311 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈
ℝ*) |
| 28 | | nnuz 12921 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
| 29 | | 1zzd 12648 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1
∈ ℤ) |
| 30 | | fveq2 6906 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
| 31 | 30 | ineq2d 4220 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘))) |
| 32 | 31 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ (𝐹‘𝑘)))) |
| 33 | | voliunlem.6 |
. . . . . . . . . . . . . . 15
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛)))) |
| 34 | | fvex 6919 |
. . . . . . . . . . . . . . 15
⊢
(vol*‘(𝑥 ∩
(𝐹‘𝑘))) ∈ V |
| 35 | 32, 33, 34 | fvmpt 7016 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘)))) |
| 36 | 35 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘)))) |
| 37 | | inss1 4237 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 |
| 38 | | ovolsscl 25521 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
| 39 | 37, 38 | mp3an1 1450 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
| 40 | 39 | 3adant1 1131 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
| 41 | 40 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
| 42 | 36, 41 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ) |
| 43 | 28, 29, 42 | serfre 14072 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( +
, 𝐻):ℕ⟶ℝ) |
| 44 | 43 | frnd 6744 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran
seq1( + , 𝐻) ⊆
ℝ) |
| 45 | | ressxr 11305 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℝ* |
| 46 | 44, 45 | sstrdi 3996 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran
seq1( + , 𝐻) ⊆
ℝ*) |
| 47 | | supxrcl 13357 |
. . . . . . . . 9
⊢ (ran
seq1( + , 𝐻) ⊆
ℝ* → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈
ℝ*) |
| 48 | 46, 47 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran
seq1( + , 𝐻),
ℝ*, < ) ∈ ℝ*) |
| 49 | | simp3 1139 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘𝑥) ∈
ℝ) |
| 50 | 49, 23 | resubcld 11691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘𝑥) −
(vol*‘(𝑥 ∖
∪ ran 𝐹))) ∈ ℝ) |
| 51 | 50 | rexrd 11311 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘𝑥) −
(vol*‘(𝑥 ∖
∪ ran 𝐹))) ∈
ℝ*) |
| 52 | | iunin2 5071 |
. . . . . . . . . . 11
⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪
𝑛 ∈ ℕ (𝐹‘𝑛)) |
| 53 | | ffn 6736 |
. . . . . . . . . . . . . 14
⊢ (𝐹:ℕ⟶dom vol →
𝐹 Fn
ℕ) |
| 54 | | fniunfv 7267 |
. . . . . . . . . . . . . 14
⊢ (𝐹 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
| 55 | 1, 53, 54 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
| 56 | 55 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
| 57 | 56 | ineq2d 4220 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹)) |
| 58 | 52, 57 | eqtrid 2789 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹)) |
| 59 | 58 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹))) |
| 60 | | eqid 2737 |
. . . . . . . . . 10
⊢ seq1( + ,
𝐻) = seq1( + , 𝐻) |
| 61 | | inss1 4237 |
. . . . . . . . . . . 12
⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 |
| 62 | 61, 14 | sstrid 3995 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ) |
| 63 | 62 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ) |
| 64 | | ovolsscl 25521 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
| 65 | 61, 64 | mp3an1 1450 |
. . . . . . . . . . . 12
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
| 66 | 65 | 3adant1 1131 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
| 67 | 66 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
| 68 | 60, 33, 63, 67 | ovoliun 25540 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, <
)) |
| 69 | 59, 68 | eqbrtrrd 5167 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, <
)) |
| 70 | 1 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom
vol) |
| 71 | | voliunlem.5 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖)) |
| 72 | 71 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
Disj 𝑖 ∈
ℕ (𝐹‘𝑖)) |
| 73 | 70, 72, 33, 14, 49 | voliunlem1 25585 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥)) |
| 74 | 43 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( +
, 𝐻)‘𝑘) ∈
ℝ) |
| 75 | 23 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) →
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ) |
| 76 | | simpl3 1194 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) →
(vol*‘𝑥) ∈
ℝ) |
| 77 | | leaddsub 11739 |
. . . . . . . . . . . . 13
⊢ (((seq1(
+ , 𝐻)‘𝑘) ∈ ℝ ∧
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 78 | 74, 75, 76, 77 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((seq1(
+ , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 79 | 73, 78 | mpbid 232 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( +
, 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))) |
| 80 | 79 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∀𝑘 ∈ ℕ
(seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))) |
| 81 | | ffn 6736 |
. . . . . . . . . . 11
⊢ (seq1( +
, 𝐻):ℕ⟶ℝ
→ seq1( + , 𝐻) Fn
ℕ) |
| 82 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ (seq1( + ,
𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 83 | 82 | ralrn 7108 |
. . . . . . . . . . 11
⊢ (seq1( +
, 𝐻) Fn ℕ →
(∀𝑧 ∈ ran seq1(
+ , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 84 | 43, 81, 83 | 3syl 18 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(∀𝑧 ∈ ran seq1(
+ , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 85 | 80, 84 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∀𝑧 ∈ ran seq1(
+ , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 86 | | supxrleub 13368 |
. . . . . . . . . 10
⊢ ((ran
seq1( + , 𝐻) ⊆
ℝ* ∧ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ∈
ℝ*) → (sup(ran seq1( + , 𝐻), ℝ*, < ) ≤
((vol*‘𝑥) −
(vol*‘(𝑥 ∖
∪ ran 𝐹))) ↔ ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 87 | 46, 51, 86 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(sup(ran seq1( + , 𝐻),
ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 88 | 85, 87 | mpbird 257 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran
seq1( + , 𝐻),
ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 89 | 27, 48, 51, 69, 88 | xrletrd 13204 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 90 | | leaddsub 11739 |
. . . . . . . 8
⊢
(((vol*‘(𝑥
∩ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(((vol*‘(𝑥 ∩
∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 91 | 18, 23, 49, 90 | syl3anc 1373 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(((vol*‘(𝑥 ∩
∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 92 | 89, 91 | mpbird 257 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥)) |
| 93 | 18, 23 | readdcld 11290 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ∈
ℝ) |
| 94 | 49, 93 | letri3d 11403 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔
((vol*‘𝑥) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ∧
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥)))) |
| 95 | 26, 92, 94 | mpbir2and 713 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
| 96 | 95 | 3expia 1122 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ →
(vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 97 | 10, 96 | sylan2 593 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ℝ) →
((vol*‘𝑥) ∈
ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))) |
| 98 | 97 | ralrimiva 3146 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ →
(vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
| 99 | | ismbl 25561 |
. 2
⊢ (∪ ran 𝐹 ∈ dom vol ↔ (∪ ran 𝐹 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫
ℝ((vol*‘𝑥)
∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))) |
| 100 | 9, 98, 99 | sylanbrc 583 |
1
⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol) |