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Theorem voliunlem1 25585
Description: Lemma for voliun 25589. (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 4136 . . . 4 (𝐸 ran 𝐹) ⊆ 𝐸
2 voliunlem1.7 . . . 4 (𝜑𝐸 ⊆ ℝ)
3 voliunlem1.8 . . . . 5 (𝜑 → (vol*‘𝐸) ∈ ℝ)
43adantr 480 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5 ovolsscl 25521 . . . 4 (((𝐸 ran 𝐹) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
61, 2, 4, 5mp3an2ani 1470 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ran 𝐹)) ∈ ℝ)
7 difss 4136 . . . 4 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
8 ovolsscl 25521 . . . 4 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
97, 2, 4, 8mp3an2ani 1470 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
10 inss1 4237 . . . 4 (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸
11 ovolsscl 25521 . . . 4 (((𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
1210, 2, 4, 11mp3an2ani 1470 . . 3 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℝ)
13 elfznn 13593 . . . . . . . . 9 (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
14 voliunlem.3 . . . . . . . . . . . 12 (𝜑𝐹:ℕ⟶dom vol)
1514ffnd 6737 . . . . . . . . . . 11 (𝜑𝐹 Fn ℕ)
16 fnfvelrn 7100 . . . . . . . . . . 11 ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
1715, 16sylan 580 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ ran 𝐹)
18 elssuni 4937 . . . . . . . . . 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 3146 . . . . . . 7 (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2221adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
23 iunss 5045 . . . . . 6 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2422, 23sylibr 234 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ⊆ ran 𝐹)
2524sscond 4146 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 ran 𝐹) ⊆ (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
262adantr 480 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
277, 26sstrid 3995 . . . 4 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)) ⊆ ℝ)
28 ovolss 25520 . . . 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 11878 . 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 7439 . . . . . . . . . . 11 (𝑧 = 1 → (1...𝑧) = (1...1))
3231iuneq1d 5019 . . . . . . . . . 10 (𝑧 = 1 → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...1)(𝐹𝑛))
3332eleq1d 2826 . . . . . . . . 9 (𝑧 = 1 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol))
3432ineq2d 4220 . . . . . . . . . . 11 (𝑧 = 1 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)))
3534fveq2d 6910 . . . . . . . . . 10 (𝑧 = 1 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))))
36 fveq2 6906 . . . . . . . . . 10 (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
3735, 36eqeq12d 2753 . . . . . . . . 9 (𝑧 = 1 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
3833, 37anbi12d 632 . . . . . . . 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 340 . . . . . . 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 7439 . . . . . . . . . . 11 (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
4140iuneq1d 5019 . . . . . . . . . 10 (𝑧 = 𝑘 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
4241eleq1d 2826 . . . . . . . . 9 (𝑧 = 𝑘 → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol))
4341ineq2d 4220 . . . . . . . . . . 11 (𝑧 = 𝑘 → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
4443fveq2d 6910 . . . . . . . . . 10 (𝑧 = 𝑘 → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
45 fveq2 6906 . . . . . . . . . 10 (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
4644, 45eqeq12d 2753 . . . . . . . . 9 (𝑧 = 𝑘 → ((vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
4742, 46anbi12d 632 . . . . . . . 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 340 . . . . . . 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 7439 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
5049iuneq1d 5019 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → 𝑛 ∈ (1...𝑧)(𝐹𝑛) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
5150eleq1d 2826 . . . . . . . . 9 (𝑧 = (𝑘 + 1) → ( 𝑛 ∈ (1...𝑧)(𝐹𝑛) ∈ dom vol ↔ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
5250ineq2d 4220 . . . . . . . . . . 11 (𝑧 = (𝑘 + 1) → (𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛)) = (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)))
5352fveq2d 6910 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 𝑛 ∈ (1...𝑧)(𝐹𝑛))) = (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))))
54 fveq2 6906 . . . . . . . . . 10 (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
5553, 54eqeq12d 2753 . . . . . . . . 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 632 . . . . . . . 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 340 . . . . . . 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 12647 . . . . . . . . . . 11 1 ∈ ℤ
59 fzsn 13606 . . . . . . . . . . 11 (1 ∈ ℤ → (1...1) = {1})
60 iuneq1 5008 . . . . . . . . . . 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 11257 . . . . . . . . . . 11 1 ∈ V
63 fveq2 6906 . . . . . . . . . . 11 (𝑛 = 1 → (𝐹𝑛) = (𝐹‘1))
6462, 63iunxsn 5091 . . . . . . . . . 10 𝑛 ∈ {1} (𝐹𝑛) = (𝐹‘1)
6561, 64eqtri 2765 . . . . . . . . 9 𝑛 ∈ (1...1)(𝐹𝑛) = (𝐹‘1)
66 1nn 12277 . . . . . . . . . 10 1 ∈ ℕ
67 ffvelcdm 7101 . . . . . . . . . 10 ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
6814, 66, 67sylancl 586 . . . . . . . . 9 (𝜑 → (𝐹‘1) ∈ dom vol)
6965, 68eqeltrid 2845 . . . . . . . 8 (𝜑 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol)
7063ineq2d 4220 . . . . . . . . . . . 12 (𝑛 = 1 → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘1)))
7170fveq2d 6910 . . . . . . . . . . 11 (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))))
72 voliunlem1.6 . . . . . . . . . . 11 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹𝑛))))
73 fvex 6919 . . . . . . . . . . 11 (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
7471, 72, 73fvmpt 7016 . . . . . . . . . 10 (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
7566, 74ax-mp 5 . . . . . . . . 9 (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
76 seq1 14055 . . . . . . . . . 10 (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
7758, 76ax-mp 5 . . . . . . . . 9 (seq1( + , 𝐻)‘1) = (𝐻‘1)
7865ineq2i 4217 . . . . . . . . . 10 (𝐸 𝑛 ∈ (1...1)(𝐹𝑛)) = (𝐸 ∩ (𝐹‘1))
7978fveq2i 6909 . . . . . . . . 9 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))
8075, 77, 793eqtr4ri 2776 . . . . . . . 8 (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)
8169, 80jctir 520 . . . . . . 7 (𝜑 → ( 𝑛 ∈ (1...1)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...1)(𝐹𝑛))) = (seq1( + , 𝐻)‘1)))
82 peano2nn 12278 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83 ffvelcdm 7101 . . . . . . . . . . . . 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 25572 . . . . . . . . . . . . 13 (( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)
8685ex 412 . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
89 nnuz 12921 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
9088, 89eleqtrdi 2851 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ‘1))
91 fzsuc 13611 . . . . . . . . . . . . . 14 (𝑘 ∈ (ℤ‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
92 iuneq1 5008 . . . . . . . . . . . . . 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 5094 . . . . . . . . . . . . . 14 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛))
95 ovex 7464 . . . . . . . . . . . . . . . 16 (𝑘 + 1) ∈ V
96 fveq2 6906 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (𝐹𝑛) = (𝐹‘(𝑘 + 1)))
9795, 96iunxsn 5091 . . . . . . . . . . . . . . 15 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛) = (𝐹‘(𝑘 + 1))
9897uneq2i 4165 . . . . . . . . . . . . . 14 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹𝑛)) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
9994, 98eqtri 2765 . . . . . . . . . . . . 13 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1)))
10093, 99eqtrdi 2793 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) = ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))))
101100eleq1d 2826 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol ↔ ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
10287, 101sylibrd 259 . . . . . . . . . 10 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol → 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∈ dom vol))
103 oveq1 7438 . . . . . . . . . . 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 4237 . . . . . . . . . . . . . . 15 (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸
105104, 26sstrid 3995 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ ℝ)
106 ovolsscl 25521 . . . . . . . . . . . . . . 15 (((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
107104, 2, 4, 106mp3an2ani 1470 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) ∈ ℝ)
108 mblsplit 25567 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 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 4230 . . . . . . . . . . . . . . . 16 ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
111 inss2 4238 . . . . . . . . . . . . . . . . . 18 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
11282adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
113112, 89eleqtrdi 2851 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ‘1))
114 eluzfz2 13572 . . . . . . . . . . . . . . . . . . 19 ((𝑘 + 1) ∈ (ℤ‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
11596ssiun2s 5048 . . . . . . . . . . . . . . . . . . 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 3995 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
118 dfss2 3969 . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
120110, 119eqtrid 2789 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
121120fveq2d 6910 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
122 indif2 4281 . . . . . . . . . . . . . . . 16 (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))
123 uncom 4158 . . . . . . . . . . . . . . . . . . 19 ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
124100, 123eqtr2di 2794 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))
125 voliunlem.5 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑Disj 𝑖 ∈ ℕ (𝐹𝑖))
126125ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹𝑖))
127112adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
12813adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
129128nnred 12281 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
130 elfzle2 13568 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ (1...𝑘) → 𝑛𝑘)
131130adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛𝑘)
13288adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
133 nnleltp1 12673 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛𝑘𝑛 < (𝑘 + 1)))
134128, 132, 133syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛𝑘𝑛 < (𝑘 + 1)))
135131, 134mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
136129, 135gtned 11396 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
137 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = (𝑘 + 1) → (𝐹𝑖) = (𝐹‘(𝑘 + 1)))
138 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑖 = 𝑛 → (𝐹𝑖) = (𝐹𝑛))
139137, 138disji2 5127 . . . . . . . . . . . . . . . . . . . . . 22 ((Disj 𝑖 ∈ ℕ (𝐹𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
140126, 127, 128, 136, 139syl121anc 1377 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ∅)
141140iuneq2dv 5016 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = 𝑛 ∈ (1...𝑘)∅)
142 iunin2 5071 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛))
143 iun0 5062 . . . . . . . . . . . . . . . . . . . 20 𝑛 ∈ (1...𝑘)∅ = ∅
144141, 142, 1433eqtr3g 2800 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ 𝑛 ∈ (1...𝑘)(𝐹𝑛)) = ∅)
145 uneqdifeq 4493 . . . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1))) = 𝑛 ∈ (1...𝑘)(𝐹𝑛))
148147ineq2d 4220 . . . . . . . . . . . . . . . 16 ((𝜑𝑘 ∈ ℕ) → (𝐸 ∩ ( 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
149122, 148eqtr3id 2791 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → ((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))
150149fveq2d 6910 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘((𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
151121, 150oveq12d 7449 . . . . . . . . . . . . 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 4237 . . . . . . . . . . . . . . . 16 (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
153 ovolsscl 25521 . . . . . . . . . . . . . . . 16 (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
154152, 2, 4, 153mp3an2ani 1470 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
155154recnd 11289 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
15612recnd 11289 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) ∈ ℂ)
157155, 156addcomd 11463 . . . . . . . . . . . . 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 2781 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...(𝑘 + 1))(𝐹𝑛))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
159 seqp1 14057 . . . . . . . . . . . . . 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 4220 . . . . . . . . . . . . . . . . 17 (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
162161fveq2d 6910 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
163 fvex 6919 . . . . . . . . . . . . . . . 16 (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
164162, 72, 163fvmpt 7016 . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
167160, 166eqtrd 2777 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
168158, 167eqeq12d 2753 . . . . . . . . . . 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, 168imbitrrid 246 . . . . . . . . . 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 413 . . . . . . . 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 12284 . . . . . 6 (𝑘 ∈ ℕ → (𝜑 → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))))
174173impcom 407 . . . . 5 ((𝜑𝑘 ∈ ℕ) → ( 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol ∧ (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘)))
175174simprd 495 . . . 4 ((𝜑𝑘 ∈ ℕ) → (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) = (seq1( + , 𝐻)‘𝑘))
176175eqcomd 2743 . . 3 ((𝜑𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))))
177176oveq1d 7446 . 2 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 ran 𝐹))))
178174simpld 494 . . 3 ((𝜑𝑘 ∈ ℕ) → 𝑛 ∈ (1...𝑘)(𝐹𝑛) ∈ dom vol)
179 mblsplit 25567 . . 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 1373 . 2 ((𝜑𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛))) + (vol*‘(𝐸 𝑛 ∈ (1...𝑘)(𝐹𝑛)))))
18130, 177, 1803brtr4d 5175 1 ((𝜑𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ran 𝐹))) ≤ (vol*‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626   cuni 4907   ciun 4991  Disj wdisj 5110   class class class wbr 5143  cmpt 5225  dom cdm 5685  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cr 11154  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cn 12266  cz 12613  cuz 12878  ...cfz 13547  seqcseq 14042  vol*covol 25497  volcvol 25498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fl 13832  df-seq 14043  df-exp 14103  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-ovol 25499  df-vol 25500
This theorem is referenced by:  voliunlem2  25586  voliunlem3  25587
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