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Theorem voliunlem1 25533

Description: Lemma for voliun 25537. (Contributed by Mario Carneiro, 20-Mar-2014.)

**Hypotheses**
Ref	Expression
voliunlem.3	⊢ (𝜑 → 𝐹:ℕ⟶dom vol)
voliunlem.5	⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
voliunlem1.6	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹‘𝑛))))
voliunlem1.7	⊢ (𝜑 → 𝐸 ⊆ ℝ)
voliunlem1.8	⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)

**Assertion**
Ref	Expression
voliunlem1	⊢ ((𝜑 ∧ 𝑘 ∈ 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.

Step	Hyp	Ref	Expression
1		difss 4077	. . . 4 ⊢ (𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸
2		voliunlem1.7	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
3		voliunlem1.8	. . . . 5 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
4	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5		ovolsscl 25469	. . . 4 ⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∖ ∪ ran 𝐹)) ∈ ℝ)
6	1, 2, 4, 5	mp3an2ani 1471	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ∈ ℝ)
7		difss 4077	. . . 4 ⊢ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸
8		ovolsscl 25469	. . . 4 ⊢ (((𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ)
9	7, 2, 4, 8	mp3an2ani 1471	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ)
10		inss1 4178	. . . 4 ⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸
11		ovolsscl 25469	. . . 4 ⊢ (((𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ)
12	10, 2, 4, 11	mp3an2ani 1471	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ)
13		elfznn 13504	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
14		voliunlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶dom vol)
15	14	ffnd 6667	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn ℕ)
16		fnfvelrn 7030	. . . . . . . . . . 11 ⊢ ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
17	15, 16	sylan 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
18		elssuni 4882	. . . . . . . . . 10 ⊢ ((𝐹‘𝑛) ∈ ran 𝐹 → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
20	13, 19	sylan2 594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
21	20	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
23		iunss 4988	. . . . . 6 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
24	22, 23	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
25	24	sscond 4087	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪ ran 𝐹) ⊆ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
26	2	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
27	7, 26	sstrid 3934	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ)
28		ovolss 25468	. . . 4 ⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ∧ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ) → (vol‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
29	25, 27, 28	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
30	6, 9, 12, 29	leadd2dd 11762	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol‘(𝐸 ∖ ∪ ran 𝐹))) ≤ ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))))
31		oveq2 7372	. . . . . . . . . . 11 ⊢ (𝑧 = 1 → (1...𝑧) = (1...1))
32	31	iuneq1d 4962	. . . . . . . . . 10 ⊢ (𝑧 = 1 → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))
33	32	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑧 = 1 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol))
34	32	ineq2d 4161	. . . . . . . . . . 11 ⊢ (𝑧 = 1 → (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)))
35	34	fveq2d 6842	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))))
36		fveq2 6838	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
37	35, 36	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑧 = 1 → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)))
38	33, 37	anbi12d 633	. . . . . . . 8 ⊢ (𝑧 = 1 → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))))
39	38	imbi2d 340	. . . . . . 7 ⊢ (𝑧 = 1 → ((𝜑 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)))))
40		oveq2 7372	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
41	40	iuneq1d 4962	. . . . . . . . . 10 ⊢ (𝑧 = 𝑘 → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
42	41	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑧 = 𝑘 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol))
43	41	ineq2d 4161	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑘 → (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
44	43	fveq2d 6842	. . . . . . . . . 10 ⊢ (𝑧 = 𝑘 → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
45		fveq2 6838	. . . . . . . . . 10 ⊢ (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
46	44, 45	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑧 = 𝑘 → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))
47	42, 46	anbi12d 633	. . . . . . . 8 ⊢ (𝑧 = 𝑘 → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))))
48	47	imbi2d 340	. . . . . . 7 ⊢ (𝑧 = 𝑘 → ((𝜑 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))))
49		oveq2 7372	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
50	49	iuneq1d 4962	. . . . . . . . . 10 ⊢ (𝑧 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
51	50	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑧 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol))
52	50	ineq2d 4161	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑘 + 1) → (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)))
53	52	fveq2d 6842	. . . . . . . . . 10 ⊢ (𝑧 = (𝑘 + 1) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))))
54		fveq2 6838	. . . . . . . . . 10 ⊢ (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
55	53, 54	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑧 = (𝑘 + 1) → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
56	51, 55	anbi12d 633	. . . . . . . 8 ⊢ (𝑧 = (𝑘 + 1) → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))
57	56	imbi2d 340	. . . . . . 7 ⊢ (𝑧 = (𝑘 + 1) → ((𝜑 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))))
58		1z 12554	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
59		fzsn 13517	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (1...1) = {1})
60		iuneq1 4951	. . . . . . . . . . 11 ⊢ ((1...1) = {1} → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛))
61	58, 59, 60	mp2b 10	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛)
62		1ex 11137	. . . . . . . . . . 11 ⊢ 1 ∈ V
63		fveq2 6838	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
64	62, 63	iunxsn 5034	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1)
65	61, 64	eqtri 2760	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)
66		1nn 12182	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
67		ffvelcdm 7031	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
68	14, 66, 67	sylancl 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ dom vol)
69	65, 68	eqeltrid 2841	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol)
70	63	ineq2d 4161	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1)))
71	70	fveq2d 6842	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (vol‘(𝐸 ∩ (𝐹‘𝑛))) = (vol‘(𝐸 ∩ (𝐹‘1))))
72		voliunlem1.6	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹‘𝑛))))
73		fvex 6851	. . . . . . . . . . 11 ⊢ (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
74	71, 72, 73	fvmpt 6945	. . . . . . . . . 10 ⊢ (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
75	66, 74	ax-mp 5	. . . . . . . . 9 ⊢ (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
76		seq1 13973	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
77	58, 76	ax-mp 5	. . . . . . . . 9 ⊢ (seq1( + , 𝐻)‘1) = (𝐻‘1)
78	65	ineq2i 4158	. . . . . . . . . 10 ⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1))
79	78	fveq2i 6841	. . . . . . . . 9 ⊢ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ (𝐹‘1)))
80	75, 77, 79	3eqtr4ri 2771	. . . . . . . 8 ⊢ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)
81	69, 80	jctir 520	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)))
82		peano2nn 12183	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈ ℕ)
83		ffvelcdm 7031	. . . . . . . . . . . . 13 ⊢ ((𝐹:ℕ⟶dom vol ∧ (𝑘 + 1) ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
84	14, 82, 83	syl2an 597	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
85		unmbl 25520	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)
86	85	ex 412	. . . . . . . . . . . 12 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ((𝐹‘(𝑘 + 1)) ∈ dom vol → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
87	84, 86	syl5com 31	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
88		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
89		nnuz 12824	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
90	88, 89	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ_≥‘1))
91		fzsuc 13522	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}))
92		iuneq1 4951	. . . . . . . . . . . . . 14 ⊢ ((1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)}) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛))
93	90, 91, 92	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛))
94		iunxun 5037	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛))
95		ovex 7397	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 + 1) ∈ V
96		fveq2 6838	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
97	95, 96	iunxsn 5034	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))
98	97	uneq2i 4106	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
99	94, 98	eqtri 2760	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
100	93, 99	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))))
101	100	eleq1d 2822	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ↔ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol))
102	87, 101	sylibrd 259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol))
103		oveq1 7371	. . . . . . . . . . 11 ⊢ ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘) → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
104		inss1 4178	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸
105	104, 26	sstrid 3934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ)
106		ovolsscl 25469	. . . . . . . . . . . . . . 15 ⊢ (((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ)
107	104, 2, 4, 106	mp3an2ani 1471	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ)
108		mblsplit 25515	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘(𝑘 + 1)) ∈ dom vol ∧ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))))))
109	84, 105, 107, 108	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))))))
110		in32 4171	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
111		inss2 4179	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
112	82	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
113	112, 89	eleqtrdi 2847	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ_≥‘1))
114		eluzfz2 13483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
115	96	ssiun2s 4992	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 + 1) ∈ (1...(𝑘 + 1)) → (𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
116	113, 114, 115	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
117	111, 116	sstrid 3934	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
118		dfss2 3908	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
119	117, 118	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
120	110, 119	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
121	120	fveq2d 6842	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
122		indif2 4222	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∩ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))
123		uncom 4099	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
124	100, 123	eqtr2di 2789	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
125		voliunlem.5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
126	125	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
127	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
128	13	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
129	128	nnred 12186	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
130		elfzle2 13479	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ≤ 𝑘)
131	130	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ≤ 𝑘)
132	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
133		nnleltp1 12581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1)))
134	128, 132, 133	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1)))
135	131, 134	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
136	129, 135	gtned 11278	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
137		fveq2 6838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = (𝑘 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑘 + 1)))
138		fveq2 6838	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
139	137, 138	disji2 5070	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Disj 𝑖 ∈ ℕ (𝐹‘𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅)
140	126, 127, 128, 136, 139	syl121anc 1378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅)
141	140	iuneq2dv 4959	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∪ 𝑛 ∈ (1...𝑘)∅)
142		iunin2 5014	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
143		iun0 5005	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑛 ∈ (1...𝑘)∅ = ∅
144	141, 142, 143	3eqtr3g 2795	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅)
145		uneqdifeq 4433	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∧ ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅) → (((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
146	116, 144, 145	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
147	124, 146	mpbid 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
148	147	ineq2d 4161	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
149	122, 148	eqtr3id 2786	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
150	149	fveq2d 6842	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
151	121, 150	oveq12d 7382	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))))) = ((vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))))
152		inss1 4178	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
153		ovolsscl 25469	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
154	152, 2, 4, 153	mp3an2ani 1471	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
155	154	recnd 11170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
156	12	recnd 11170	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℂ)
157	155, 156	addcomd 11345	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) = ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
158	109, 151, 157	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
159		seqp1 13975	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (ℤ_≥‘1) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
160	90, 159	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))))
161	96	ineq2d 4161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
162	161	fveq2d 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (vol‘(𝐸 ∩ (𝐹‘𝑛))) = (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
163		fvex 6851	. . . . . . . . . . . . . . . 16 ⊢ (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
164	162, 72, 163	fvmpt 6945	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 + 1) ∈ ℕ → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
165	112, 164	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
166	165	oveq2d 7380	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
167	160, 166	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
168	158, 167	eqeq12d 2753	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)) ↔ ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))))
169	103, 168	imbitrrid 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))
170	102, 169	anim12d 610	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))
171	170	expcom 413	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ → (𝜑 → ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))))
172	171	a2d 29	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → ((𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) → (𝜑 → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))))
173	39, 48, 57, 48, 81, 172	nnind 12189	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))))
174	173	impcom 407	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))
175	174	simprd 495	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))
176	175	eqcomd 2743	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
177	176	oveq1d 7379	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol‘(𝐸 ∖ ∪ ran 𝐹))) = ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪ ran 𝐹))))
178	174	simpld 494	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol)
179		mblsplit 25515	. . 3 ⊢ ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘𝐸) = ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))))
180	178, 26, 4, 179	syl3anc 1374	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘𝐸) = ((vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))))
181	30, 177, 180	3brtr4d 5118	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 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 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∖ cdif 3887 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4274 {csn 4568 ∪ cuni 4851 ∪ ciun 4934 Disj wdisj 5053 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5628 ran crn 5629 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 (class class class)co 7364 ℝcr 11034 1c1 11036 + caddc 11038 < clt 11176 ≤ cle 11177 ℕcn 12171 ℤcz 12521 ℤ_≥cuz 12785 ...cfz 13458 seqcseq 13960 vol*covol 25445 volcvol 25446

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5306 ax-pr 5374 ax-un 7686 ax-cnex 11091 ax-resscn 11092 ax-1cn 11093 ax-icn 11094 ax-addcl 11095 ax-addrcl 11096 ax-mulcl 11097 ax-mulrcl 11098 ax-mulcom 11099 ax-addass 11100 ax-mulass 11101 ax-distr 11102 ax-i2m1 11103 ax-1ne0 11104 ax-1rid 11105 ax-rnegex 11106 ax-rrecex 11107 ax-cnre 11108 ax-pre-lttri 11109 ax-pre-lttrn 11110 ax-pre-ltadd 11111 ax-pre-mulgt0 11112 ax-pre-sup 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5523 df-eprel 5528 df-po 5536 df-so 5537 df-fr 5581 df-we 5583 df-xp 5634 df-rel 5635 df-cnv 5636 df-co 5637 df-dm 5638 df-rn 5639 df-res 5640 df-ima 5641 df-pred 6263 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7321 df-ov 7367 df-oprab 7368 df-mpo 7369 df-om 7815 df-1st 7939 df-2nd 7940 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9352 df-inf 9353 df-pnf 11178 df-mnf 11179 df-xr 11180 df-ltxr 11181 df-le 11182 df-sub 11376 df-neg 11377 df-div 11805 df-nn 12172 df-2 12241 df-3 12242 df-n0 12435 df-z 12522 df-uz 12786 df-q 12896 df-rp 12940 df-ioo 13299 df-ico 13301 df-icc 13302 df-fz 13459 df-fl 13748 df-seq 13961 df-exp 14021 df-cj 15058 df-re 15059 df-im 15060 df-sqrt 15194 df-abs 15195 df-ovol 25447 df-vol 25448

This theorem is referenced by: voliunlem2 25534 voliunlem3 25535

W3C validator