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Theorem voliunlem1 24724
Description: Lemma for voliun 24728. (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 difss 4065 . . . 4 (𝐸 ran 𝐹) ⊆ 𝐸
2 voliunlem1.7 . . . 4 (𝜑𝐸 ⊆ ℝ)
3 voliunlem1.8 . . . . 5 (𝜑 → (vol*‘𝐸) ∈ ℝ)
43adantr 481 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5 ovolsscl 24660 . . . 4 (((𝐸 ran 𝐹) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
61, 2, 4, 5mp3an2ani 1467 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
7 difss 4065 . . . 4 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
8 ovolsscl 24660 . . . 4 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
97, 2, 4, 8mp3an2ani 1467 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
10 inss1 4162 . . . 4 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
11 ovolsscl 24660 . . . 4 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
1210, 2, 4, 11mp3an2ani 1467 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
13 elfznn 13295 . . . . . . . . 9 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
14 voliunlem.3 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶dom vol)
1514ffnd 6593 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
16 fnfvelrn 6950 . . . . . . . . . . 11 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
1715, 16sylan 580 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
18 elssuni 4871 . . . . . . . . . 10 ((𝐹𝑛) ∈ ran 𝐹 → (𝐹𝑛) ⊆ ran 𝐹)
1917, 18syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ⊆ ran 𝐹)
2013, 19sylan2 593 . . . . . . . 8 ((𝜑𝑛 ∈ (1...𝑘)) → (𝐹𝑛) ⊆ ran 𝐹)
2120ralrimiva 3108 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2221adantr 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
23 iunss 4974 . . . . . 6 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2422, 23sylibr 233 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2524sscond 4075 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 ran 𝐹) ⊆ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
262adantr 481 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
277, 26sstrid 3931 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ ℝ)
28 ovolss 24659 . . . 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))))
2925, 27, 28syl2anc 584 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ≤ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
306, 9, 12, 29leadd2dd 11600 . 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)))))
31 oveq2 7275 . . . . . . . . . . 11 (𝑧 = 1 → (1...𝑧) = (1...1))
3231iuneq1d 4951 . . . . . . . . . 10 (𝑧 = 1 → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3332eleq1d 2823 . . . . . . . . 9 (𝑧 = 1 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol))
3432ineq2d 4146 . . . . . . . . . . 11 (𝑧 = 1 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)))
3534fveq2d 6770 . . . . . . . . . 10 (𝑧 = 1 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))))
36 fveq2 6766 . . . . . . . . . 10 (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
3735, 36eqeq12d 2754 . . . . . . . . 9 (𝑧 = 1 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
3833, 37anbi12d 631 . . . . . . . 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))))
3938imbi2d 341 . . . . . . 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)))))
40 oveq2 7275 . . . . . . . . . . 11 (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
4140iuneq1d 4951 . . . . . . . . . 10 (𝑧 = 𝑘 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
4241eleq1d 2823 . . . . . . . . 9 (𝑧 = 𝑘 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol))
4341ineq2d 4146 . . . . . . . . . . 11 (𝑧 = 𝑘 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
4443fveq2d 6770 . . . . . . . . . 10 (𝑧 = 𝑘 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
45 fveq2 6766 . . . . . . . . . 10 (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
4644, 45eqeq12d 2754 . . . . . . . . 9 (𝑧 = 𝑘 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
4742, 46anbi12d 631 . . . . . . . 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))))
4847imbi2d 341 . . . . . . 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)))))
49 oveq2 7275 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
5049iuneq1d 4951 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
5150eleq1d 2823 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
5250ineq2d 4146 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)))
5352fveq2d 6770 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))))
54 fveq2 6766 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
5553, 54eqeq12d 2754 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
5651, 55anbi12d 631 . . . . . . . 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)))))
5756imbi2d 341 . . . . . . 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))))))
58 1z 12360 . . . . . . . . . . 11 1 ∈ ℤ
59 fzsn 13308 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
60 iuneq1 4940 . . . . . . . . . . 11 ((1...1) = {1} → 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛))
6158, 59, 60mp2b 10 . . . . . . . . . 10 𝑛 ∈ (1...1)(𝐹𝑛) = 𝑛 ∈ {1} (𝐹𝑛)
62 1ex 10981 . . . . . . . . . . 11 1 ∈ V
63 fveq2 6766 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
6462, 63iunxsn 5019 . . . . . . . . . 10 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
6561, 64eqtri 2766 . . . . . . . . 9 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
66 1nn 11994 . . . . . . . . . 10 1 ∈ ℕ
67 ffvelrn 6951 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
6814, 66, 67sylancl 586 . . . . . . . . 9 (𝜑 → (𝐹‘1) ∈ dom vol)
6965, 68eqeltrid 2843 . . . . . . . 8 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol)
7063ineq2d 4146 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘1)))
7170fveq2d 6770 . . . . . . . . . . 11 (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))))
72 voliunlem1.6 . . . . . . . . . . 11 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))
73 fvex 6779 . . . . . . . . . . 11 (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
7471, 72, 73fvmpt 6867 . . . . . . . . . 10 (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
7566, 74ax-mp 5 . . . . . . . . 9 (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
76 seq1 13744 . . . . . . . . . 10 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
7758, 76ax-mp 5 . . . . . . . . 9 (seq1( + , 𝐻)‘1) = (𝐻‘1)
7865ineq2i 4143 . . . . . . . . . 10 (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)) = (𝐸 ∩ (𝐹‘1))
7978fveq2i 6769 . . . . . . . . 9 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))
8075, 77, 793eqtr4ri 2777 . . . . . . . 8 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)
8169, 80jctir 521 . . . . . . 7 (𝜑 → ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
82 peano2nn 11995 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83 ffvelrn 6951 . . . . . . . . . . . . 13 ((𝐹:ℕ⟶dom vol ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
8414, 82, 83syl2an 596 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
85 unmbl 24711 . . . . . . . . . . . . 13 (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)
8685ex 413 . . . . . . . . . . . 12 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → ((𝐹‘(𝑘 + 1)) ∈ dom vol → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
8784, 86syl5com 31 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
88 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
89 nnuz 12631 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
9088, 89eleqtrdi 2849 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
91 fzsuc 13313 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
92 iuneq1 4940 . . . . . . . . . . . . . 14 ((1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
9390, 91, 923syl 18 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛))
94 iunxun 5022 . . . . . . . . . . . . . 14 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
95 ovex 7300 . . . . . . . . . . . . . . . 16 (𝑘 + 1) ∈ V
96 fveq2 6766 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
9795, 96iunxsn 5019 . . . . . . . . . . . . . . 15 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
9897uneq2i 4093 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
9994, 98eqtri 2766 . . . . . . . . . . . . 13 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10093, 99eqtrdi 2794 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
101100eleq1d 2823 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ↔ ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
10287, 101sylibrd 258 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
103 oveq1 7274 . . . . . . . . . . 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))))))
104 inss1 4162 . . . . . . . . . . . . . . 15 (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸
105104, 26sstrid 3931 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ ℝ)
106 ovolsscl 24660 . . . . . . . . . . . . . . 15 (((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
107104, 2, 4, 106mp3an2ani 1467 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
108 mblsplit 24706 . . . . . . . . . . . . . 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))))))
10984, 105, 107, 108syl3anc 1370 . . . . . . . . . . . . 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))))))
110 in32 4155 . . . . . . . . . . . . . . . 16 ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
111 inss2 4163 . . . . . . . . . . . . . . . . . 18 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
11282adantl 482 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
113112, 89eleqtrdi 2849 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
114 eluzfz2 13274 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (ℤ‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
11596ssiun2s 4977 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (1...(𝑘 + 1)) → (𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
116113, 114, 1153syl 18 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
117111, 116sstrid 3931 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
118 df-ss 3903 . . . . . . . . . . . . . . . . 17 ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ↔ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
119117, 118sylib 217 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
120110, 119eqtrid 2790 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
121120fveq2d 6770 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
122 indif2 4204 . . . . . . . . . . . . . . . 16 (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))
123 uncom 4086 . . . . . . . . . . . . . . . . . . 19 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
124100, 123eqtr2di 2795 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
125 voliunlem.5 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
126125ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
127112adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
12813adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
129128nnred 11998 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
130 elfzle2 13270 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑘) → 𝑛𝑘)
131130adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛𝑘)
13288adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
133 nnleltp1 12385 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛𝑘𝑛 < (𝑘 + 1)))
134128, 132, 133syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛𝑘𝑛 < (𝑘 + 1)))
135131, 134mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
136129, 135gtned 11120 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
137 fveq2 6766 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = (𝑘 + 1) → (𝐹𝑖) = (𝐹‘(𝑘 + 1)))
138 fveq2 6766 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
139137, 138disji2 5055 . . . . . . . . . . . . . . . . . . . . . 22 ((Disj 𝑖 ∈ ℕ (𝐹𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
140126, 127, 128, 136, 139syl121anc 1374 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
141140iuneq2dv 4948 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = 𝑛 ∈ (1...𝑘)∅)
142 iunin2 4999 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
143 iun0 4990 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)∅ = ∅
144141, 142, 1433eqtr3g 2801 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = ∅)
145 uneqdifeq 4423 . . . . . . . . . . . . . . . . . . 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)))
146116, 144, 145syl2anc 584 . . . . . . . . . . . . . . . . . 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)))
147124, 146mpbid 231 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1))) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
148147ineq2d 4146 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
149122, 148eqtr3id 2792 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
150149fveq2d 6770 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
151121, 150oveq12d 7285 . . . . . . . . . . . . 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)))))
152 inss1 4162 . . . . . . . . . . . . . . . 16 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
153 ovolsscl 24660 . . . . . . . . . . . . . . . 16 (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
154152, 2, 4, 153mp3an2ani 1467 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
155154recnd 11013 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
15612recnd 11013 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℂ)
157155, 156addcomd 11187 . . . . . . . . . . . . 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))))))
158109, 151, 1573eqtrd 2782 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
159 seqp1 13746 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
16090, 159syl 17 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
16196ineq2d 4146 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
162161fveq2d 6770 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
163 fvex 6779 . . . . . . . . . . . . . . . 16 (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
164162, 72, 163fvmpt 6867 . . . . . . . . . . . . . . 15 ((𝑘 + 1) ∈ ℕ → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
165112, 164syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
166165oveq2d 7283 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
167160, 166eqtrd 2778 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
168158, 167eqeq12d 2754 . . . . . . . . . . 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)))))))
169103, 168syl5ibr 245 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
170102, 169anim12d 609 . . . . . . . . 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)))))
171170expcom 414 . . . . . . . 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))))))
172171a2d 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))))))
17339, 48, 57, 48, 81, 172nnind 12001 . . . . . 6 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))))
174173impcom 408 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
175174simprd 496 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))
176175eqcomd 2744 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
177176oveq1d 7282 . 2 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ran 𝐹))))
178174simpld 495 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol)
179 mblsplit 24706 . . 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)))))
180178, 26, 4, 179syl3anc 1370 . 2 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
18130, 177, 1803brtr4d 5105 1 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  cdif 3883  cun 3884  cin 3885  wss 3886  c0 4256  {csn 4561   cuni 4839   ciun 4924  Disj wdisj 5038   class class class wbr 5073  cmpt 5156  dom cdm 5584  ran crn 5585   Fn wfn 6421  wf 6422  cfv 6426  (class class class)co 7267  cr 10880  1c1 10882   + caddc 10884   < clt 11019  cle 11020  cn 11983  cz 12329  cuz 12592  ...cfz 13249  seqcseq 13731  vol*covol 24636  volcvol 24637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pow 5286  ax-pr 5350  ax-un 7578  ax-cnex 10937  ax-resscn 10938  ax-1cn 10939  ax-icn 10940  ax-addcl 10941  ax-addrcl 10942  ax-mulcl 10943  ax-mulrcl 10944  ax-mulcom 10945  ax-addass 10946  ax-mulass 10947  ax-distr 10948  ax-i2m1 10949  ax-1ne0 10950  ax-1rid 10951  ax-rnegex 10952  ax-rrecex 10953  ax-cnre 10954  ax-pre-lttri 10955  ax-pre-lttrn 10956  ax-pre-ltadd 10957  ax-pre-mulgt0 10958  ax-pre-sup 10959
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3431  df-sbc 3716  df-csb 3832  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-pss 3905  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-disj 5039  df-br 5074  df-opab 5136  df-mpt 5157  df-tr 5191  df-id 5484  df-eprel 5490  df-po 5498  df-so 5499  df-fr 5539  df-we 5541  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-rn 5595  df-res 5596  df-ima 5597  df-pred 6195  df-ord 6262  df-on 6263  df-lim 6264  df-suc 6265  df-iota 6384  df-fun 6428  df-fn 6429  df-f 6430  df-f1 6431  df-fo 6432  df-f1o 6433  df-fv 6434  df-riota 7224  df-ov 7270  df-oprab 7271  df-mpo 7272  df-om 7703  df-1st 7820  df-2nd 7821  df-frecs 8084  df-wrecs 8115  df-recs 8189  df-rdg 8228  df-er 8485  df-map 8604  df-en 8721  df-dom 8722  df-sdom 8723  df-sup 9188  df-inf 9189  df-pnf 11021  df-mnf 11022  df-xr 11023  df-ltxr 11024  df-le 11025  df-sub 11217  df-neg 11218  df-div 11643  df-nn 11984  df-2 12046  df-3 12047  df-n0 12244  df-z 12330  df-uz 12593  df-q 12699  df-rp 12741  df-ioo 13093  df-ico 13095  df-icc 13096  df-fz 13250  df-fl 13522  df-seq 13732  df-exp 13793  df-cj 14820  df-re 14821  df-im 14822  df-sqrt 14956  df-abs 14957  df-ovol 24638  df-vol 24639
This theorem is referenced by:  voliunlem2  24725  voliunlem3  24726
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