| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | difss 4135 | . . . 4
⊢ (𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸 | 
| 2 |  | voliunlem1.7 | . . . 4
⊢ (𝜑 → 𝐸 ⊆ ℝ) | 
| 3 |  | voliunlem1.8 | . . . . 5
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) | 
| 4 | 3 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘𝐸) ∈
ℝ) | 
| 5 |  | ovolsscl 25522 | . . . 4
⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
∪ ran 𝐹)) ∈ ℝ) | 
| 6 | 1, 2, 4, 5 | mp3an2ani 1469 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ∈ ℝ) | 
| 7 |  | difss 4135 | . . . 4
⊢ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 | 
| 8 |  | ovolsscl 25522 | . . . 4
⊢ (((𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) | 
| 9 | 7, 2, 4, 8 | mp3an2ani 1469 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) | 
| 10 |  | inss1 4236 | . . . 4
⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 | 
| 11 |  | ovolsscl 25522 | . . . 4
⊢ (((𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) | 
| 12 | 10, 2, 4, 11 | mp3an2ani 1469 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) | 
| 13 |  | elfznn 13594 | . . . . . . . . 9
⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ) | 
| 14 |  | voliunlem.3 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:ℕ⟶dom vol) | 
| 15 | 14 | ffnd 6736 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn ℕ) | 
| 16 |  | fnfvelrn 7099 | . . . . . . . . . . 11
⊢ ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹) | 
| 17 | 15, 16 | sylan 580 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹) | 
| 18 |  | elssuni 4936 | . . . . . . . . . 10
⊢ ((𝐹‘𝑛) ∈ ran 𝐹 → (𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 19 | 17, 18 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 20 | 13, 19 | sylan2 593 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 21 | 20 | ralrimiva 3145 | . . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 22 | 21 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 23 |  | iunss 5044 | . . . . . 6
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 24 | 22, 23 | sylibr 234 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) | 
| 25 | 24 | sscond 4145 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪ ran
𝐹) ⊆ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) | 
| 26 | 2 | adantr 480 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ) | 
| 27 | 7, 26 | sstrid 3994 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ) | 
| 28 |  | ovolss 25521 | . . . 4
⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ (𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ∧ (𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) | 
| 29 | 25, 27, 28 | syl2anc 584 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) | 
| 30 | 6, 9, 12, 29 | leadd2dd 11879 | . 2
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪ ran
𝐹))) ≤
((vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) | 
| 31 |  | oveq2 7440 | . . . . . . . . . . 11
⊢ (𝑧 = 1 → (1...𝑧) = (1...1)) | 
| 32 | 31 | iuneq1d 5018 | . . . . . . . . . 10
⊢ (𝑧 = 1 → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)) | 
| 33 | 32 | eleq1d 2825 | . . . . . . . . 9
⊢ (𝑧 = 1 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol)) | 
| 34 | 32 | ineq2d 4219 | . . . . . . . . . . 11
⊢ (𝑧 = 1 → (𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪
𝑛 ∈ (1...1)(𝐹‘𝑛))) | 
| 35 | 34 | fveq2d 6909 | . . . . . . . . . 10
⊢ (𝑧 = 1 → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...1)(𝐹‘𝑛)))) | 
| 36 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1)) | 
| 37 | 35, 36 | eqeq12d 2752 | . . . . . . . . 9
⊢ (𝑧 = 1 → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))) | 
| 38 | 33, 37 | anbi12d 632 | . . . . . . . 8
⊢ (𝑧 = 1 → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)))) | 
| 39 | 38 | imbi2d 340 | . . . . . . 7
⊢ (𝑧 = 1 → ((𝜑 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪
𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))))) | 
| 40 |  | oveq2 7440 | . . . . . . . . . . 11
⊢ (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘)) | 
| 41 | 40 | iuneq1d 5018 | . . . . . . . . . 10
⊢ (𝑧 = 𝑘 → ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) | 
| 42 | 41 | eleq1d 2825 | . . . . . . . . 9
⊢ (𝑧 = 𝑘 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol)) | 
| 43 | 41 | ineq2d 4219 | . . . . . . . . . . 11
⊢ (𝑧 = 𝑘 → (𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) | 
| 44 | 43 | fveq2d 6909 | . . . . . . . . . 10
⊢ (𝑧 = 𝑘 → (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) | 
| 45 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘)) | 
| 46 | 44, 45 | eqeq12d 2752 | . . . . . . . . 9
⊢ (𝑧 = 𝑘 → ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) | 
| 47 | 42, 46 | anbi12d 632 | . . . . . . . 8
⊢ (𝑧 = 𝑘 → ((∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))) | 
| 48 | 47 | imbi2d 340 | . . . . . . 7
⊢ (𝑧 = 𝑘 → ((𝜑 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))))) | 
| 49 |  | oveq2 7440 | . . . . . . . . . . 11
⊢ (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1))) | 
| 50 | 49 | iuneq1d 5018 | . . . . . . . . . 10
⊢ (𝑧 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) | 
| 51 | 50 | eleq1d 2825 | . . . . . . . . 9
⊢ (𝑧 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol)) | 
| 52 | 50 | ineq2d 4219 | . . . . . . . . . . 11
⊢ (𝑧 = (𝑘 + 1) → (𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) | 
| 53 | 52 | fveq2d 6909 | . . . . . . . . . 10
⊢ (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)))) | 
| 54 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1))) | 
| 55 | 53, 54 | eqeq12d 2752 | . . . . . . . . 9
⊢ (𝑧 = (𝑘 + 1) → ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))) | 
| 56 | 51, 55 | anbi12d 632 | . . . . . . . 8
⊢ (𝑧 = (𝑘 + 1) → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))) | 
| 57 | 56 | imbi2d 340 | . . . . . . 7
⊢ (𝑧 = (𝑘 + 1) → ((𝜑 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))) | 
| 58 |  | 1z 12649 | . . . . . . . . . . 11
⊢ 1 ∈
ℤ | 
| 59 |  | fzsn 13607 | . . . . . . . . . . 11
⊢ (1 ∈
ℤ → (1...1) = {1}) | 
| 60 |  | iuneq1 5007 | . . . . . . . . . . 11
⊢ ((1...1)
= {1} → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛)) | 
| 61 | 58, 59, 60 | mp2b 10 | . . . . . . . . . 10
⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛) | 
| 62 |  | 1ex 11258 | . . . . . . . . . . 11
⊢ 1 ∈
V | 
| 63 |  | fveq2 6905 | . . . . . . . . . . 11
⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) | 
| 64 | 62, 63 | iunxsn 5090 | . . . . . . . . . 10
⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1) | 
| 65 | 61, 64 | eqtri 2764 | . . . . . . . . 9
⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1) | 
| 66 |  | 1nn 12278 | . . . . . . . . . 10
⊢ 1 ∈
ℕ | 
| 67 |  | ffvelcdm 7100 | . . . . . . . . . 10
⊢ ((𝐹:ℕ⟶dom vol ∧ 1
∈ ℕ) → (𝐹‘1) ∈ dom vol) | 
| 68 | 14, 66, 67 | sylancl 586 | . . . . . . . . 9
⊢ (𝜑 → (𝐹‘1) ∈ dom vol) | 
| 69 | 65, 68 | eqeltrid 2844 | . . . . . . . 8
⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol) | 
| 70 | 63 | ineq2d 4219 | . . . . . . . . . . . 12
⊢ (𝑛 = 1 → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1))) | 
| 71 | 70 | fveq2d 6909 | . . . . . . . . . . 11
⊢ (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹‘𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))) | 
| 72 |  | voliunlem1.6 | . . . . . . . . . . 11
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹‘𝑛)))) | 
| 73 |  | fvex 6918 | . . . . . . . . . . 11
⊢
(vol*‘(𝐸 ∩
(𝐹‘1))) ∈
V | 
| 74 | 71, 72, 73 | fvmpt 7015 | . . . . . . . . . 10
⊢ (1 ∈
ℕ → (𝐻‘1)
= (vol*‘(𝐸 ∩
(𝐹‘1)))) | 
| 75 | 66, 74 | ax-mp 5 | . . . . . . . . 9
⊢ (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))) | 
| 76 |  | seq1 14056 | . . . . . . . . . 10
⊢ (1 ∈
ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1)) | 
| 77 | 58, 76 | ax-mp 5 | . . . . . . . . 9
⊢ (seq1( +
, 𝐻)‘1) = (𝐻‘1) | 
| 78 | 65 | ineq2i 4216 | . . . . . . . . . 10
⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1)) | 
| 79 | 78 | fveq2i 6908 | . . . . . . . . 9
⊢
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))) | 
| 80 | 75, 77, 79 | 3eqtr4ri 2775 | . . . . . . . 8
⊢
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1) | 
| 81 | 69, 80 | jctir 520 | . . . . . . 7
⊢ (𝜑 → (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))) | 
| 82 |  | peano2nn 12279 | . . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) | 
| 83 |  | ffvelcdm 7100 | . . . . . . . . . . . . 13
⊢ ((𝐹:ℕ⟶dom vol ∧
(𝑘 + 1) ∈ ℕ)
→ (𝐹‘(𝑘 + 1)) ∈ dom
vol) | 
| 84 | 14, 82, 83 | syl2an 596 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol) | 
| 85 |  | unmbl 25573 | . . . . . . . . . . . . 13
⊢
((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol) | 
| 86 | 85 | ex 412 | . . . . . . . . . . . 12
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ((𝐹‘(𝑘 + 1)) ∈ dom vol → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)) | 
| 87 | 84, 86 | syl5com 31 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)) | 
| 88 |  | simpr 484 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | 
| 89 |  | nnuz 12922 | . . . . . . . . . . . . . . 15
⊢ ℕ =
(ℤ≥‘1) | 
| 90 | 88, 89 | eleqtrdi 2850 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈
(ℤ≥‘1)) | 
| 91 |  | fzsuc 13612 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈
(ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) | 
| 92 |  | iuneq1 5007 | . . . . . . . . . . . . . 14
⊢
((1...(𝑘 + 1)) =
((1...𝑘) ∪ {(𝑘 + 1)}) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛)) | 
| 93 | 90, 91, 92 | 3syl 18 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛)) | 
| 94 |  | iunxun 5093 | . . . . . . . . . . . . . 14
⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪
𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) | 
| 95 |  | ovex 7465 | . . . . . . . . . . . . . . . 16
⊢ (𝑘 + 1) ∈ V | 
| 96 |  | fveq2 6905 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))) | 
| 97 | 95, 96 | iunxsn 5090 | . . . . . . . . . . . . . . 15
⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)) | 
| 98 | 97 | uneq2i 4164 | . . . . . . . . . . . . . 14
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪
𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) | 
| 99 | 94, 98 | eqtri 2764 | . . . . . . . . . . . . 13
⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) | 
| 100 | 93, 99 | eqtrdi 2792 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))) | 
| 101 | 100 | eleq1d 2825 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ↔ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)) | 
| 102 | 87, 101 | sylibrd 259 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol)) | 
| 103 |  | oveq1 7439 | . . . . . . . . . . 11
⊢
((vol*‘(𝐸
∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘) → ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) | 
| 104 |  | inss1 4236 | . . . . . . . . . . . . . . 15
⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸 | 
| 105 | 104, 26 | sstrid 3994 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ) | 
| 106 |  | ovolsscl 25522 | . . . . . . . . . . . . . . 15
⊢ (((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) | 
| 107 | 104, 2, 4, 106 | mp3an2ani 1469 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) | 
| 108 |  | mblsplit 25568 | . . . . . . . . . . . . . 14
⊢ (((𝐹‘(𝑘 + 1)) ∈ dom vol ∧ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))))) | 
| 109 | 84, 105, 107, 108 | syl3anc 1372 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))))) | 
| 110 |  | in32 4229 | . . . . . . . . . . . . . . . 16
⊢ ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) | 
| 111 |  | inss2 4237 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1)) | 
| 112 | 82 | adantl 481 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) | 
| 113 | 112, 89 | eleqtrdi 2850 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈
(ℤ≥‘1)) | 
| 114 |  | eluzfz2 13573 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 + 1) ∈
(ℤ≥‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1))) | 
| 115 | 96 | ssiun2s 5047 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 + 1) ∈ (1...(𝑘 + 1)) → (𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) | 
| 116 | 113, 114,
115 | 3syl 18 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) | 
| 117 | 111, 116 | sstrid 3994 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) | 
| 118 |  | dfss2 3968 | . . . . . . . . . . . . . . . . 17
⊢ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) | 
| 119 | 117, 118 | sylib 218 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) | 
| 120 | 110, 119 | eqtrid 2788 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) | 
| 121 | 120 | fveq2d 6909 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) | 
| 122 |  | indif2 4280 | . . . . . . . . . . . . . . . 16
⊢ (𝐸 ∩ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))) | 
| 123 |  | uncom 4157 | . . . . . . . . . . . . . . . . . . 19
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) | 
| 124 | 100, 123 | eqtr2di 2793 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) | 
| 125 |  | voliunlem.5 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖)) | 
| 126 | 125 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖)) | 
| 127 | 112 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ) | 
| 128 | 13 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ) | 
| 129 | 128 | nnred 12282 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ) | 
| 130 |  | elfzle2 13569 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ≤ 𝑘) | 
| 131 | 130 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ≤ 𝑘) | 
| 132 | 88 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ) | 
| 133 |  | nnleltp1 12675 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1))) | 
| 134 | 128, 132,
133 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1))) | 
| 135 | 131, 134 | mpbid 232 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1)) | 
| 136 | 129, 135 | gtned 11397 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛) | 
| 137 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 = (𝑘 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑘 + 1))) | 
| 138 |  | fveq2 6905 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛)) | 
| 139 | 137, 138 | disji2 5126 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((Disj 𝑖
∈ ℕ (𝐹‘𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅) | 
| 140 | 126, 127,
128, 136, 139 | syl121anc 1376 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅) | 
| 141 | 140 | iuneq2dv 5015 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∪
𝑛 ∈ (1...𝑘)∅) | 
| 142 |  | iunin2 5070 | . . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) | 
| 143 |  | iun0 5061 | . . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑛 ∈ (1...𝑘)∅ = ∅ | 
| 144 | 141, 142,
143 | 3eqtr3g 2799 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅) | 
| 145 |  | uneqdifeq 4492 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∧ ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅) → (((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) | 
| 146 | 116, 144,
145 | syl2anc 584 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) | 
| 147 | 124, 146 | mpbid 232 | . . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) | 
| 148 | 147 | ineq2d 4219 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) | 
| 149 | 122, 148 | eqtr3id 2790 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) | 
| 150 | 149 | fveq2d 6909 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) | 
| 151 | 121, 150 | oveq12d 7450 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))))) = ((vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) | 
| 152 |  | inss1 4236 | . . . . . . . . . . . . . . . 16
⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸 | 
| 153 |  | ovolsscl 25522 | . . . . . . . . . . . . . . . 16
⊢ (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ) | 
| 154 | 152, 2, 4, 153 | mp3an2ani 1469 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ) | 
| 155 | 154 | recnd 11290 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ) | 
| 156 | 12 | recnd 11290 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℂ) | 
| 157 | 155, 156 | addcomd 11464 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) = ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) | 
| 158 | 109, 151,
157 | 3eqtrd 2780 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) | 
| 159 |  | seqp1 14058 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈
(ℤ≥‘1) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1)))) | 
| 160 | 90, 159 | syl 17 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1)))) | 
| 161 | 96 | ineq2d 4219 | . . . . . . . . . . . . . . . . 17
⊢ (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) | 
| 162 | 161 | fveq2d 6909 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹‘𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) | 
| 163 |  | fvex 6918 | . . . . . . . . . . . . . . . 16
⊢
(vol*‘(𝐸 ∩
(𝐹‘(𝑘 + 1)))) ∈
V | 
| 164 | 162, 72, 163 | fvmpt 7015 | . . . . . . . . . . . . . . 15
⊢ ((𝑘 + 1) ∈ ℕ →
(𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) | 
| 165 | 112, 164 | syl 17 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) | 
| 166 | 165 | oveq2d 7448 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) | 
| 167 | 160, 166 | eqtrd 2776 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) | 
| 168 | 158, 167 | eqeq12d 2752 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)) ↔ ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))) | 
| 169 | 103, 168 | imbitrrid 246 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘) → (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))) | 
| 170 | 102, 169 | anim12d 609 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))) | 
| 171 | 170 | expcom 413 | . . . . . . . 8
⊢ (𝑘 ∈ ℕ → (𝜑 → ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))) | 
| 172 | 171 | a2d 29 | . . . . . . 7
⊢ (𝑘 ∈ ℕ → ((𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) → (𝜑 → (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))) | 
| 173 | 39, 48, 57, 48, 81, 172 | nnind 12285 | . . . . . 6
⊢ (𝑘 ∈ ℕ → (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))) | 
| 174 | 173 | impcom 407 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) | 
| 175 | 174 | simprd 495 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) | 
| 176 | 175 | eqcomd 2742 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) | 
| 177 | 176 | oveq1d 7447 | . 2
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∖ ∪ ran
𝐹))) = ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪ ran
𝐹)))) | 
| 178 | 174 | simpld 494 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol) | 
| 179 |  | mblsplit 25568 | . . 3
⊢
((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘𝐸) =
((vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) | 
| 180 | 178, 26, 4, 179 | syl3anc 1372 | . 2
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) | 
| 181 | 30, 177, 180 | 3brtr4d 5174 | 1
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝐸)) |