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Theorem voliunlem1 23241
 Description: Lemma for voliun 23245. (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 481 . . . 4 ((𝜑𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
3 voliunlem1.8 . . . . 5 (𝜑 → (vol*‘𝐸) ∈ ℝ)
43adantr 481 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5 difss 3720 . . . . 5 (𝐸 ran 𝐹) ⊆ 𝐸
6 ovolsscl 23177 . . . . 5 (((𝐸 ran 𝐹) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
75, 6mp3an1 1408 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
82, 4, 7syl2anc 692 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
9 difss 3720 . . . . 5 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
10 ovolsscl 23177 . . . . 5 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
119, 10mp3an1 1408 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
122, 4, 11syl2anc 692 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
13 inss1 3816 . . . . 5 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
14 ovolsscl 23177 . . . . 5 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
1513, 14mp3an1 1408 . . . 4 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
162, 4, 15syl2anc 692 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
17 elfznn 12320 . . . . . . . . 9 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
18 voliunlem.3 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶dom vol)
19 ffn 6007 . . . . . . . . . . . 12 (𝐹:ℕ⟶dom vol → 𝐹 Fn ℕ)
2018, 19syl 17 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
21 fnfvelrn 6317 . . . . . . . . . . 11 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
2220, 21sylan 488 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
23 elssuni 4438 . . . . . . . . . 10 ((𝐹𝑛) ∈ ran 𝐹 → (𝐹𝑛) ⊆ ran 𝐹)
2422, 23syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
2517, 24sylan2 491 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝑘)) → (𝐹𝑛) ⊆ ran 𝐹)
2625ralrimiva 2961 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2726adantr 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
28 iunss 4532 . . . . . 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 3730 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 ran 𝐹) ⊆ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
319, 2syl5ss 3598 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ ℝ)
32 ovolss 23176 . . . 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 692 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ≤ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
348, 12, 16, 33leadd2dd 10594 . 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 6618 . . . . . . . . . . 11 (𝑧 = 1 → (1...𝑧) = (1...1))
3635iuneq1d 4516 . . . . . . . . . 10 (𝑧 = 1 → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3736eleq1d 2683 . . . . . . . . 9 (𝑧 = 1 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol))
3836ineq2d 3797 . . . . . . . . . . 11 (𝑧 = 1 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)))
3938fveq2d 6157 . . . . . . . . . 10 (𝑧 = 1 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))))
40 fveq2 6153 . . . . . . . . . 10 (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
4139, 40eqeq12d 2636 . . . . . . . . 9 (𝑧 = 1 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
4237, 41anbi12d 746 . . . . . . . 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 330 . . . . . . 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 6618 . . . . . . . . . . 11 (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
4544iuneq1d 4516 . . . . . . . . . 10 (𝑧 = 𝑘 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
4645eleq1d 2683 . . . . . . . . 9 (𝑧 = 𝑘 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol))
4745ineq2d 3797 . . . . . . . . . . 11 (𝑧 = 𝑘 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
4847fveq2d 6157 . . . . . . . . . 10 (𝑧 = 𝑘 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
49 fveq2 6153 . . . . . . . . . 10 (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
5048, 49eqeq12d 2636 . . . . . . . . 9 (𝑧 = 𝑘 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
5146, 50anbi12d 746 . . . . . . . 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 330 . . . . . . 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 6618 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
5453iuneq1d 4516 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
5554eleq1d 2683 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
5654ineq2d 3797 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)))
5756fveq2d 6157 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))))
58 fveq2 6153 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
5957, 58eqeq12d 2636 . . . . . . . . 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 746 . . . . . . . 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 330 . . . . . . 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 11359 . . . . . . . . . . 11 1 ∈ ℤ
63 fzsn 12333 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
64 iuneq1 4505 . . . . . . . . . . 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 9987 . . . . . . . . . . 11 1 ∈ V
67 fveq2 6153 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
6866, 67iunxsn 4574 . . . . . . . . . 10 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
6965, 68eqtri 2643 . . . . . . . . 9 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
70 1nn 10983 . . . . . . . . . 10 1 ∈ ℕ
71 ffvelrn 6318 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
7218, 70, 71sylancl 693 . . . . . . . . 9 (𝜑 → (𝐹‘1) ∈ dom vol)
7369, 72syl5eqel 2702 . . . . . . . 8 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol)
7467ineq2d 3797 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘1)))
7574fveq2d 6157 . . . . . . . . . . 11 (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))))
76 voliunlem1.6 . . . . . . . . . . 11 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))
77 fvex 6163 . . . . . . . . . . 11 (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
7875, 76, 77fvmpt 6244 . . . . . . . . . 10 (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
7970, 78ax-mp 5 . . . . . . . . 9 (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
80 seq1 12762 . . . . . . . . . 10 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
8162, 80ax-mp 5 . . . . . . . . 9 (seq1( + , 𝐻)‘1) = (𝐻‘1)
8269ineq2i 3794 . . . . . . . . . 10 (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)) = (𝐸 ∩ (𝐹‘1))
8382fveq2i 6156 . . . . . . . . 9 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))
8479, 81, 833eqtr4ri 2654 . . . . . . . 8 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)
8573, 84jctir 560 . . . . . . 7 (𝜑 → ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
86 peano2nn 10984 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
87 ffvelrn 6318 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
8818, 86, 87syl2an 494 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
89 unmbl 23228 . . . . . . . . . . . . 13 (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)
9089ex 450 . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
93 nnuz 11675 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
9492, 93syl6eleq 2708 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
95 fzsuc 12338 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
96 iuneq1 4505 . . . . . . . . . . . . . 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 4576 . . . . . . . . . . . . . 14 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
99 ovex 6638 . . . . . . . . . . . . . . . 16 (𝑘 + 1) ∈ V
100 fveq2 6153 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
10199, 100iunxsn 4574 . . . . . . . . . . . . . . 15 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
102101uneq2i 3747 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10398, 102eqtri 2643 . . . . . . . . . . . . 13 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10497, 103syl6eq 2671 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
105104eleq1d 2683 . . . . . . . . . . 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 6617 . . . . . . . . . . 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 3816 . . . . . . . . . . . . . . 15 (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸
109108, 2syl5ss 3598 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ ℝ)
110 ovolsscl 23177 . . . . . . . . . . . . . . . 16 (((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
111108, 110mp3an1 1408 . . . . . . . . . . . . . . 15 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
1122, 4, 111syl2anc 692 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
113 mblsplit 23223 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . 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 3808 . . . . . . . . . . . . . . . 16 ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
116 inss2 3817 . . . . . . . . . . . . . . . . . 18 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
11786adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
118117, 93syl6eleq 2708 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
119 eluzfz2 12299 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (ℤ‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
120100ssiun2s 4535 . . . . . . . . . . . . . . . . . . 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 3598 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
123 df-ss 3573 . . . . . . . . . . . . . . . . 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 2667 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
126125fveq2d 6157 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
127 indif2 3851 . . . . . . . . . . . . . . . 16 (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))
128 uncom 3740 . . . . . . . . . . . . . . . . . . 19 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
129104, 128syl6req 2672 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
130 voliunlem.5 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
131130ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
132117adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
13317adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
134133nnred 10987 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
135 elfzle2 12295 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑘) → 𝑛𝑘)
136135adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛𝑘)
13792adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
138 nnleltp1 11384 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛𝑘𝑛 < (𝑘 + 1)))
139133, 137, 138syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛𝑘𝑛 < (𝑘 + 1)))
140136, 139mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
141134, 140gtned 10124 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
142 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = (𝑘 + 1) → (𝐹𝑖) = (𝐹‘(𝑘 + 1)))
143 fveq2 6153 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
144142, 143disji2 4604 . . . . . . . . . . . . . . . . . . . . . 22 ((Disj 𝑖 ∈ ℕ (𝐹𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
145131, 132, 133, 141, 144syl121anc 1328 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
146145iuneq2dv 4513 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = 𝑛 ∈ (1...𝑘)∅)
147 iunin2 4555 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
148 iun0 4547 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)∅ = ∅
149146, 147, 1483eqtr3g 2678 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = ∅)
150 uneqdifeq 4034 . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . 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 3797 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
154127, 153syl5eqr 2669 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
155154fveq2d 6157 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
156126, 155oveq12d 6628 . . . . . . . . . . . . 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 3816 . . . . . . . . . . . . . . . . 17 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
158 ovolsscl 23177 . . . . . . . . . . . . . . . . 17 (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
159157, 158mp3an1 1408 . . . . . . . . . . . . . . . 16 ((𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
1602, 4, 159syl2anc 692 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
161160recnd 10020 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
16216recnd 10020 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℂ)
163161, 162addcomd 10190 . . . . . . . . . . . . 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 2659 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
165 seqp1 12764 . . . . . . . . . . . . . 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 3797 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
168167fveq2d 6157 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
169 fvex 6163 . . . . . . . . . . . . . . . 16 (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
170168, 76, 169fvmpt 6244 . . . . . . . . . . . . . . 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 6626 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
173166, 172eqtrd 2655 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
174164, 173eqeq12d 2636 . . . . . . . . . . 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 585 . . . . . . . . 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 451 . . . . . . . 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 10990 . . . . . 6 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))))
180179impcom 446 . . . . 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 2627 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
183182oveq1d 6625 . 2 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ran 𝐹))))
184180simpld 475 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol)
185 mblsplit 23223 . . 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 1323 . 2 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
18734, 183, 1863brtr4d 4650 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 384   = wceq 1480   ∈ wcel 1987   ≠ wne 2790  ∀wral 2907   ∖ cdif 3556   ∪ cun 3557   ∩ cin 3558   ⊆ wss 3559  ∅c0 3896  {csn 4153  ∪ cuni 4407  ∪ ciun 4490  Disj wdisj 4588   class class class wbr 4618   ↦ cmpt 4678  dom cdm 5079  ran crn 5080   Fn wfn 5847  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610  ℝcr 9887  1c1 9889   + caddc 9891   < clt 10026   ≤ cle 10027  ℕcn 10972  ℤcz 11329  ℤ≥cuz 11639  ...cfz 12276  seqcseq 12749  vol*covol 23154  volcvol 23155 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-disj 4589  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-sup 8300  df-inf 8301  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-n0 11245  df-z 11330  df-uz 11640  df-q 11741  df-rp 11785  df-ioo 12129  df-ico 12131  df-icc 12132  df-fz 12277  df-fl 12541  df-seq 12750  df-exp 12809  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-ovol 23156  df-vol 23157 This theorem is referenced by:  voliunlem2  23242  voliunlem3  23243
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