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Theorem voliunlem1 23539
Description: Lemma for voliun 23543. (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 466 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
3 voliunlem1.8 . . . . 5 (𝜑 → (vol*‘𝐸) ∈ ℝ)
43adantr 466 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5 difss 3889 . . . . 5 (𝐸 ran 𝐹) ⊆ 𝐸
6 ovolsscl 23475 . . . . 5 (((𝐸 ran 𝐹) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
75, 6mp3an1 1559 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
82, 4, 7syl2anc 567 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
9 difss 3889 . . . . 5 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
10 ovolsscl 23475 . . . . 5 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
119, 10mp3an1 1559 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
122, 4, 11syl2anc 567 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
13 inss1 3982 . . . . 5 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
14 ovolsscl 23475 . . . . 5 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
1513, 14mp3an1 1559 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
162, 4, 15syl2anc 567 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
17 elfznn 12578 . . . . . . . . 9 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
18 voliunlem.3 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶dom vol)
19 ffn 6186 . . . . . . . . . . . 12 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
2018, 19syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
21 fnfvelrn 6500 . . . . . . . . . . 11 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
2220, 21sylan 563 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
23 elssuni 4604 . . . . . . . . . 10 ((𝐹𝑛) ∈ ran 𝐹 → (𝐹𝑛) ⊆ ran 𝐹)
2422, 23syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
2517, 24sylan2 574 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝑘)) → (𝐹𝑛) ⊆ ran 𝐹)
2625ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2726adantr 466 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
28 iunss 4696 . . . . . 6 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2927, 28sylibr 224 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
3029sscond 3899 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 ran 𝐹) ⊆ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
319, 2syl5ss 3764 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ ℝ)
32 ovolss 23474 . . . 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))))
3330, 31, 32syl2anc 567 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ≤ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
348, 12, 16, 33leadd2dd 10845 . 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)))))
35 oveq2 6802 . . . . . . . . . . 11 (𝑧 = 1 → (1...𝑧) = (1...1))
3635iuneq1d 4680 . . . . . . . . . 10 (𝑧 = 1 → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3736eleq1d 2835 . . . . . . . . 9 (𝑧 = 1 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol))
3836ineq2d 3966 . . . . . . . . . . 11 (𝑧 = 1 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)))
3938fveq2d 6337 . . . . . . . . . 10 (𝑧 = 1 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))))
40 fveq2 6333 . . . . . . . . . 10 (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
4139, 40eqeq12d 2786 . . . . . . . . 9 (𝑧 = 1 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
4237, 41anbi12d 610 . . . . . . . 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))))
4342imbi2d 329 . . . . . . 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)))))
44 oveq2 6802 . . . . . . . . . . 11 (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
4544iuneq1d 4680 . . . . . . . . . 10 (𝑧 = 𝑘 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
4645eleq1d 2835 . . . . . . . . 9 (𝑧 = 𝑘 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol))
4745ineq2d 3966 . . . . . . . . . . 11 (𝑧 = 𝑘 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
4847fveq2d 6337 . . . . . . . . . 10 (𝑧 = 𝑘 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
49 fveq2 6333 . . . . . . . . . 10 (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
5048, 49eqeq12d 2786 . . . . . . . . 9 (𝑧 = 𝑘 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
5146, 50anbi12d 610 . . . . . . . 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))))
5251imbi2d 329 . . . . . . 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)))))
53 oveq2 6802 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
5453iuneq1d 4680 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
5554eleq1d 2835 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
5654ineq2d 3966 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)))
5756fveq2d 6337 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))))
58 fveq2 6333 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
5957, 58eqeq12d 2786 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
6055, 59anbi12d 610 . . . . . . . 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)))))
6160imbi2d 329 . . . . . . 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))))))
62 1z 11610 . . . . . . . . . . 11 1 ∈ ℤ
63 fzsn 12591 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
64 iuneq1 4669 . . . . . . . . . . 11 ((1...1) = {1} → 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛))
6562, 63, 64mp2b 10 . . . . . . . . . 10 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛)
66 1ex 10238 . . . . . . . . . . 11 1 ∈ V
67 fveq2 6333 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
6866, 67iunxsn 4738 . . . . . . . . . 10 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
6965, 68eqtri 2793 . . . . . . . . 9 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
70 1nn 11234 . . . . . . . . . 10 1 ∈ ℕ
71 ffvelrn 6501 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
7218, 70, 71sylancl 568 . . . . . . . . 9 (𝜑 → (𝐹‘1) ∈ dom vol)
7369, 72syl5eqel 2854 . . . . . . . 8 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol)
7467ineq2d 3966 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘1)))
7574fveq2d 6337 . . . . . . . . . . 11 (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))))
76 voliunlem1.6 . . . . . . . . . . 11 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))
77 fvex 6343 . . . . . . . . . . 11 (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
7875, 76, 77fvmpt 6425 . . . . . . . . . 10 (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
7970, 78ax-mp 5 . . . . . . . . 9 (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
80 seq1 13022 . . . . . . . . . 10 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
8162, 80ax-mp 5 . . . . . . . . 9 (seq1( + , 𝐻)‘1) = (𝐻‘1)
8269ineq2i 3963 . . . . . . . . . 10 (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)) = (𝐸 ∩ (𝐹‘1))
8382fveq2i 6336 . . . . . . . . 9 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))
8479, 81, 833eqtr4ri 2804 . . . . . . . 8 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)
8573, 84jctir 506 . . . . . . 7 (𝜑 → ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
86 peano2nn 11235 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
87 ffvelrn 6501 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
8818, 86, 87syl2an 577 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
89 unmbl 23526 . . . . . . . . . . . . 13 (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)
9089ex 397 . . . . . . . . . . . 12 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → ((𝐹‘(𝑘 + 1)) ∈ dom vol → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
9188, 90syl5com 31 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
92 simpr 471 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
93 nnuz 11926 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
9492, 93syl6eleq 2860 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
95 fzsuc 12596 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
96 iuneq1 4669 . . . . . . . . . . . . . 14 ((1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
9794, 95, 963syl 18 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
98 iunxun 4740 . . . . . . . . . . . . . 14 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
99 ovex 6824 . . . . . . . . . . . . . . . 16 (𝑘 + 1) ∈ V
100 fveq2 6333 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
10199, 100iunxsn 4738 . . . . . . . . . . . . . . 15 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
102101uneq2i 3916 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10398, 102eqtri 2793 . . . . . . . . . . . . 13 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10497, 103syl6eq 2821 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
105104eleq1d 2835 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ↔ ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
10691, 105sylibrd 249 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
107 oveq1 6801 . . . . . . . . . . 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))))))
108 inss1 3982 . . . . . . . . . . . . . . 15 (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸
109108, 2syl5ss 3764 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ ℝ)
110 ovolsscl 23475 . . . . . . . . . . . . . . . 16 (((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
111108, 110mp3an1 1559 . . . . . . . . . . . . . . 15 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
1122, 4, 111syl2anc 567 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
113 mblsplit 23521 . . . . . . . . . . . . . 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))))))
11488, 109, 112, 113syl3anc 1476 . . . . . . . . . . . . 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))))))
115 in32 3975 . . . . . . . . . . . . . . . 16 ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
116 inss2 3983 . . . . . . . . . . . . . . . . . 18 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
11786adantl 467 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
118117, 93syl6eleq 2860 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
119 eluzfz2 12557 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (ℤ‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
120100ssiun2s 4699 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (1...(𝑘 + 1)) → (𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
121118, 119, 1203syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
122116, 121syl5ss 3764 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
123 df-ss 3738 . . . . . . . . . . . . . . . . 17 ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ↔ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
124122, 123sylib 208 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
125115, 124syl5eq 2817 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
126125fveq2d 6337 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
127 indif2 4020 . . . . . . . . . . . . . . . 16 (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))
128 uncom 3909 . . . . . . . . . . . . . . . . . . 19 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
129104, 128syl6req 2822 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
130 voliunlem.5 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
131130ad2antrr 699 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
132117adantr 466 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
13317adantl 467 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
134133nnred 11238 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
135 elfzle2 12553 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑘) → 𝑛𝑘)
136135adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛𝑘)
13792adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
138 nnleltp1 11635 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛𝑘𝑛 < (𝑘 + 1)))
139133, 137, 138syl2anc 567 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛𝑘𝑛 < (𝑘 + 1)))
140136, 139mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
141134, 140gtned 10375 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
142 fveq2 6333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = (𝑘 + 1) → (𝐹𝑖) = (𝐹‘(𝑘 + 1)))
143 fveq2 6333 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
144142, 143disji2 4771 . . . . . . . . . . . . . . . . . . . . . 22 ((Disj 𝑖 ∈ ℕ (𝐹𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
145131, 132, 133, 141, 144syl121anc 1481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
146145iuneq2dv 4677 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = 𝑛 ∈ (1...𝑘)∅)
147 iunin2 4719 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
148 iun0 4711 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)∅ = ∅
149146, 147, 1483eqtr3g 2828 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = ∅)
150 uneqdifeq 4200 . . . . . . . . . . . . . . . . . . 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)))
151121, 149, 150syl2anc 567 . . . . . . . . . . . . . . . . . 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)))
152129, 151mpbid 222 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1))) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
153152ineq2d 3966 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
154127, 153syl5eqr 2819 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
155154fveq2d 6337 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
156126, 155oveq12d 6812 . . . . . . . . . . . . 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)))))
157 inss1 3982 . . . . . . . . . . . . . . . . 17 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
158 ovolsscl 23475 . . . . . . . . . . . . . . . . 17 (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
159157, 158mp3an1 1559 . . . . . . . . . . . . . . . 16 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
1602, 4, 159syl2anc 567 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
161160recnd 10271 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
16216recnd 10271 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℂ)
163161, 162addcomd 10441 . . . . . . . . . . . . 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))))))
164114, 156, 1633eqtrd 2809 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
165 seqp1 13024 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
16694, 165syl 17 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
167100ineq2d 3966 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
168167fveq2d 6337 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
169 fvex 6343 . . . . . . . . . . . . . . . 16 (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
170168, 76, 169fvmpt 6425 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ ℕ → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
171117, 170syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
172171oveq2d 6810 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
173166, 172eqtrd 2805 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
174164, 173eqeq12d 2786 . . . . . . . . . . 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)))))))
175107, 174syl5ibr 236 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
176106, 175anim12d 590 . . . . . . . . 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)))))
177176expcom 398 . . . . . . . 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))))))
178177a2d 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))))))
17943, 52, 61, 52, 85, 178nnind 11241 . . . . . 6 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))))
180179impcom 394 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
181180simprd 479 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))
182181eqcomd 2777 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
183182oveq1d 6809 . 2 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ran 𝐹))))
184180simpld 478 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol)
185 mblsplit 23521 . . 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)))))
186184, 2, 4, 185syl3anc 1476 . 2 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
18734, 183, 1863brtr4d 4819 1 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  cdif 3721  cun 3722  cin 3723  wss 3724  c0 4064  {csn 4317   cuni 4575   ciun 4655  Disj wdisj 4755   class class class wbr 4787  cmpt 4864  dom cdm 5250  ran crn 5251   Fn wfn 6027  wf 6028  cfv 6032  (class class class)co 6794  cr 10138  1c1 10140   + caddc 10142   < clt 10277  cle 10278  cn 11223  cz 11580  cuz 11889  ...cfz 12534  seqcseq 13009  vol*covol 23451  volcvol 23452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216  ax-pre-sup 10217
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-disj 4756  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-om 7214  df-1st 7316  df-2nd 7317  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-er 7897  df-map 8012  df-en 8111  df-dom 8112  df-sdom 8113  df-sup 8505  df-inf 8506  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-div 10888  df-nn 11224  df-2 11282  df-3 11283  df-n0 11496  df-z 11581  df-uz 11890  df-q 11993  df-rp 12037  df-ioo 12385  df-ico 12387  df-icc 12388  df-fz 12535  df-fl 12802  df-seq 13010  df-exp 13069  df-cj 14048  df-re 14049  df-im 14050  df-sqrt 14184  df-abs 14185  df-ovol 23453  df-vol 23454
This theorem is referenced by:  voliunlem2  23540  voliunlem3  23541
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