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Theorem voliunlem1 23537
Description: Lemma for voliun 23541. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
voliunlem.3 (𝜑𝐹:ℕ⟶dom vol)
voliunlem.5 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
voliunlem1.6 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))
voliunlem1.7 (𝜑𝐸 ⊆ ℝ)
voliunlem1.8 (𝜑 → (vol*‘𝐸) ∈ ℝ)
Assertion
Ref Expression
voliunlem1 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
Distinct variable groups:   𝑘,𝑛,𝐸   𝑖,𝑘,𝑛,𝐹   𝑘,𝐻   𝜑,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑖)   𝐸(𝑖)   𝐻(𝑖,𝑛)

Proof of Theorem voliunlem1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 voliunlem1.7 . . . . 5 (𝜑𝐸 ⊆ ℝ)
21adantr 468 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
3 voliunlem1.8 . . . . 5 (𝜑 → (vol*‘𝐸) ∈ ℝ)
43adantr 468 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5 difss 3943 . . . . 5 (𝐸 ran 𝐹) ⊆ 𝐸
6 ovolsscl 23473 . . . . 5 (((𝐸 ran 𝐹) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
75, 6mp3an1 1565 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
82, 4, 7syl2anc 575 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
9 difss 3943 . . . . 5 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
10 ovolsscl 23473 . . . . 5 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
119, 10mp3an1 1565 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
122, 4, 11syl2anc 575 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
13 inss1 4036 . . . . 5 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
14 ovolsscl 23473 . . . . 5 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
1513, 14mp3an1 1565 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
162, 4, 15syl2anc 575 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
17 elfznn 12596 . . . . . . . . 9 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
18 voliunlem.3 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶dom vol)
1918ffnd 6260 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
20 fnfvelrn 6581 . . . . . . . . . . 11 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
2119, 20sylan 571 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
22 elssuni 4668 . . . . . . . . . 10 ((𝐹𝑛) ∈ ran 𝐹 → (𝐹𝑛) ⊆ ran 𝐹)
2321, 22syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
2417, 23sylan2 582 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝑘)) → (𝐹𝑛) ⊆ ran 𝐹)
2524ralrimiva 3161 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2625adantr 468 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
27 iunss 4760 . . . . . 6 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2826, 27sylibr 225 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2928sscond 3953 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 ran 𝐹) ⊆ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
309, 2syl5ss 3816 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ ℝ)
31 ovolss 23472 . . . 4 (((𝐸 ran 𝐹) ⊆ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ∧ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ ℝ) → (vol*‘(𝐸 ran 𝐹)) ≤ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
3229, 30, 31syl2anc 575 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ≤ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
338, 12, 16, 32leadd2dd 10930 . 2 ((𝜑𝑘 ∈ ℕ) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ran 𝐹))) ≤ ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
34 oveq2 6885 . . . . . . . . . . 11 (𝑧 = 1 → (1...𝑧) = (1...1))
3534iuneq1d 4744 . . . . . . . . . 10 (𝑧 = 1 → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3635eleq1d 2877 . . . . . . . . 9 (𝑧 = 1 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol))
3735ineq2d 4020 . . . . . . . . . . 11 (𝑧 = 1 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)))
3837fveq2d 6415 . . . . . . . . . 10 (𝑧 = 1 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))))
39 fveq2 6411 . . . . . . . . . 10 (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
4038, 39eqeq12d 2828 . . . . . . . . 9 (𝑧 = 1 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
4136, 40anbi12d 618 . . . . . . . 8 (𝑧 = 1 → (( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1))))
4241imbi2d 331 . . . . . . 7 (𝑧 = 1 → ((𝜑 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))))
43 oveq2 6885 . . . . . . . . . . 11 (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
4443iuneq1d 4744 . . . . . . . . . 10 (𝑧 = 𝑘 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
4544eleq1d 2877 . . . . . . . . 9 (𝑧 = 𝑘 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol))
4644ineq2d 4020 . . . . . . . . . . 11 (𝑧 = 𝑘 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
4746fveq2d 6415 . . . . . . . . . 10 (𝑧 = 𝑘 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
48 fveq2 6411 . . . . . . . . . 10 (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
4947, 48eqeq12d 2828 . . . . . . . . 9 (𝑧 = 𝑘 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
5045, 49anbi12d 618 . . . . . . . 8 (𝑧 = 𝑘 → (( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))))
5150imbi2d 331 . . . . . . 7 (𝑧 = 𝑘 → ((𝜑 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))))
52 oveq2 6885 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
5352iuneq1d 4744 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
5453eleq1d 2877 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
5553ineq2d 4020 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)))
5655fveq2d 6415 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))))
57 fveq2 6411 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
5856, 57eqeq12d 2828 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
5954, 58anbi12d 618 . . . . . . . 8 (𝑧 = (𝑘 + 1) → (( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))
6059imbi2d 331 . . . . . . 7 (𝑧 = (𝑘 + 1) → ((𝜑 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))))
61 1z 11676 . . . . . . . . . . 11 1 ∈ ℤ
62 fzsn 12609 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
63 iuneq1 4733 . . . . . . . . . . 11 ((1...1) = {1} → 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛))
6461, 62, 63mp2b 10 . . . . . . . . . 10 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛)
65 1ex 10324 . . . . . . . . . . 11 1 ∈ V
66 fveq2 6411 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
6765, 66iunxsn 4802 . . . . . . . . . 10 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
6864, 67eqtri 2835 . . . . . . . . 9 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
69 1nn 11319 . . . . . . . . . 10 1 ∈ ℕ
70 ffvelrn 6582 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
7118, 69, 70sylancl 576 . . . . . . . . 9 (𝜑 → (𝐹‘1) ∈ dom vol)
7268, 71syl5eqel 2896 . . . . . . . 8 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol)
7366ineq2d 4020 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘1)))
7473fveq2d 6415 . . . . . . . . . . 11 (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))))
75 voliunlem1.6 . . . . . . . . . . 11 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))
76 fvex 6424 . . . . . . . . . . 11 (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
7774, 75, 76fvmpt 6506 . . . . . . . . . 10 (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
7869, 77ax-mp 5 . . . . . . . . 9 (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
79 seq1 13040 . . . . . . . . . 10 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
8061, 79ax-mp 5 . . . . . . . . 9 (seq1( + , 𝐻)‘1) = (𝐻‘1)
8168ineq2i 4017 . . . . . . . . . 10 (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)) = (𝐸 ∩ (𝐹‘1))
8281fveq2i 6414 . . . . . . . . 9 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))
8378, 80, 823eqtr4ri 2846 . . . . . . . 8 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)
8472, 83jctir 512 . . . . . . 7 (𝜑 → ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
85 peano2nn 11320 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
86 ffvelrn 6582 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
8718, 85, 86syl2an 585 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
88 unmbl 23524 . . . . . . . . . . . . 13 (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)
8988ex 399 . . . . . . . . . . . 12 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → ((𝐹‘(𝑘 + 1)) ∈ dom vol → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
9087, 89syl5com 31 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
91 simpr 473 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
92 nnuz 11944 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
9391, 92syl6eleq 2902 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
94 fzsuc 12614 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
95 iuneq1 4733 . . . . . . . . . . . . . 14 ((1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
9693, 94, 953syl 18 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
97 iunxun 4804 . . . . . . . . . . . . . 14 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
98 ovex 6909 . . . . . . . . . . . . . . . 16 (𝑘 + 1) ∈ V
99 fveq2 6411 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
10098, 99iunxsn 4802 . . . . . . . . . . . . . . 15 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
101100uneq2i 3970 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10297, 101eqtri 2835 . . . . . . . . . . . . 13 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10396, 102syl6eq 2863 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
104103eleq1d 2877 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ↔ ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
10590, 104sylibrd 250 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
106 oveq1 6884 . . . . . . . . . . 11 ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
107 inss1 4036 . . . . . . . . . . . . . . 15 (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸
108107, 2syl5ss 3816 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ ℝ)
109 ovolsscl 23473 . . . . . . . . . . . . . . . 16 (((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
110107, 109mp3an1 1565 . . . . . . . . . . . . . . 15 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
1112, 4, 110syl2anc 575 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
112 mblsplit 23519 . . . . . . . . . . . . . 14 (((𝐹‘(𝑘 + 1)) ∈ dom vol ∧ (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ ℝ ∧ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))))))
11387, 108, 111, 112syl3anc 1483 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))))))
114 in32 4029 . . . . . . . . . . . . . . . 16 ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
115 inss2 4037 . . . . . . . . . . . . . . . . . 18 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
11685adantl 469 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
117116, 92syl6eleq 2902 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
118 eluzfz2 12575 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (ℤ‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
11999ssiun2s 4763 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (1...(𝑘 + 1)) → (𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
120117, 118, 1193syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
121115, 120syl5ss 3816 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
122 df-ss 3790 . . . . . . . . . . . . . . . . 17 ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ↔ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
123121, 122sylib 209 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
124114, 123syl5eq 2859 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
125124fveq2d 6415 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
126 indif2 4079 . . . . . . . . . . . . . . . 16 (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))
127 uncom 3963 . . . . . . . . . . . . . . . . . . 19 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
128103, 127syl6req 2864 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
129 voliunlem.5 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
130129ad2antrr 708 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
131116adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
13217adantl 469 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
133132nnred 11323 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
134 elfzle2 12571 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑘) → 𝑛𝑘)
135134adantl 469 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛𝑘)
13691adantr 468 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
137 nnleltp1 11701 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛𝑘𝑛 < (𝑘 + 1)))
138132, 136, 137syl2anc 575 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛𝑘𝑛 < (𝑘 + 1)))
139135, 138mpbid 223 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
140133, 139gtned 10460 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
141 fveq2 6411 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = (𝑘 + 1) → (𝐹𝑖) = (𝐹‘(𝑘 + 1)))
142 fveq2 6411 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
143141, 142disji2 4835 . . . . . . . . . . . . . . . . . . . . . 22 ((Disj 𝑖 ∈ ℕ (𝐹𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
144130, 131, 132, 140, 143syl121anc 1487 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
145144iuneq2dv 4741 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = 𝑛 ∈ (1...𝑘)∅)
146 iunin2 4783 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
147 iun0 4775 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)∅ = ∅
148145, 146, 1473eqtr3g 2870 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = ∅)
149 uneqdifeq 4260 . . . . . . . . . . . . . . . . . . 19 (((𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∧ ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = ∅) → (((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ↔ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1))) = 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
150120, 148, 149syl2anc 575 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ↔ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1))) = 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
151128, 150mpbid 223 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1))) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
152151ineq2d 4020 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
153126, 152syl5eqr 2861 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
154153fveq2d 6415 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
155125, 154oveq12d 6895 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))))) = ((vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
156 inss1 4036 . . . . . . . . . . . . . . . . 17 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
157 ovolsscl 23473 . . . . . . . . . . . . . . . . 17 (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
158156, 157mp3an1 1565 . . . . . . . . . . . . . . . 16 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
1592, 4, 158syl2anc 575 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
160159recnd 10356 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
16116recnd 10356 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℂ)
162160, 161addcomd 10526 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
163113, 155, 1623eqtrd 2851 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
164 seqp1 13042 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
16593, 164syl 17 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
16699ineq2d 4020 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
167166fveq2d 6415 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
168 fvex 6424 . . . . . . . . . . . . . . . 16 (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
169167, 75, 168fvmpt 6506 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ ℕ → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
170116, 169syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
171170oveq2d 6893 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
172165, 171eqtrd 2847 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
173163, 172eqeq12d 2828 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)) ↔ ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))))
174106, 173syl5ibr 237 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
175105, 174anim12d 598 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))
176175expcom 400 . . . . . . . 8 (𝑘 ∈ ℕ → (𝜑 → (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))))
177176a2d 29 . . . . . . 7 (𝑘 ∈ ℕ → ((𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))) → (𝜑 → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))))
17842, 51, 60, 51, 84, 177nnind 11326 . . . . . 6 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))))
179178impcom 396 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
180179simprd 485 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))
181180eqcomd 2819 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
182181oveq1d 6892 . 2 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ran 𝐹))))
183179simpld 484 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol)
184 mblsplit 23519 . . 3 (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘𝐸) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
185183, 2, 4, 184syl3anc 1483 . 2 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
18633, 182, 1853brtr4d 4883 1 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wne 2985  wral 3103  cdif 3773  cun 3774  cin 3775  wss 3776  c0 4123  {csn 4377   cuni 4637   ciun 4719  Disj wdisj 4819   class class class wbr 4851  cmpt 4930  dom cdm 5318  ran crn 5319   Fn wfn 6099  wf 6100  cfv 6104  (class class class)co 6877  cr 10223  1c1 10225   + caddc 10227   < clt 10362  cle 10363  cn 11308  cz 11646  cuz 11907  ...cfz 12552  seqcseq 13027  vol*covol 23449  volcvol 23450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301  ax-pre-sup 10302
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-iun 4721  df-disj 4820  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-sup 8590  df-inf 8591  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-n0 11563  df-z 11647  df-uz 11908  df-q 12011  df-rp 12050  df-ioo 12400  df-ico 12402  df-icc 12403  df-fz 12553  df-fl 12820  df-seq 13028  df-exp 13087  df-cj 14065  df-re 14066  df-im 14067  df-sqrt 14201  df-abs 14202  df-ovol 23451  df-vol 23452
This theorem is referenced by:  voliunlem2  23538  voliunlem3  23539
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