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Theorem voliunlem1 24724

Description: Lemma for voliun 24728. (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 4065	. . . 4 ⊢ (𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸
2		voliunlem1.7	. . . 4 ⊢ (𝜑 → 𝐸 ⊆ ℝ)
3		voliunlem1.8	. . . . 5 ⊢ (𝜑 → (vol*‘𝐸) ∈ ℝ)
4	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘𝐸) ∈ ℝ)
5		ovolsscl 24660	. . . 4 ⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∖ ∪ ran 𝐹)) ∈ ℝ)
6	1, 2, 4, 5	mp3an2ani 1467	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ∈ ℝ)
7		difss 4065	. . . 4 ⊢ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸
8		ovolsscl 24660	. . . 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 1467	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ)
10		inss1 4162	. . . 4 ⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸
11		ovolsscl 24660	. . . 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 1467	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ)
13		elfznn 13295	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ)
14		voliunlem.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:ℕ⟶dom vol)
15	14	ffnd 6593	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn ℕ)
16		fnfvelrn 6950	. . . . . . . . . . 11 ⊢ ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
17	15, 16	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹)
18		elssuni 4871	. . . . . . . . . 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 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝐹‘𝑛) ⊆ ∪ ran 𝐹)
21	20	ralrimiva 3108	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
23		iunss 4974	. . . . . 6 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
24	22, 23	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran 𝐹)
25	24	sscond 4075	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪ ran 𝐹) ⊆ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
26	2	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ)
27	7, 26	sstrid 3931	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ)
28		ovolss 24659	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
30	6, 9, 12, 29	leadd2dd 11600	. 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 7275	. . . . . . . . . . 11 ⊢ (𝑧 = 1 → (1...𝑧) = (1...1))
32	31	iuneq1d 4951	. . . . . . . . . 10 ⊢ (𝑧 = 1 → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))
33	32	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑧 = 1 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol))
34	32	ineq2d 4146	. . . . . . . . . . 11 ⊢ (𝑧 = 1 → (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)))
35	34	fveq2d 6770	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))))
36		fveq2 6766	. . . . . . . . . 10 ⊢ (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1))
37	35, 36	eqeq12d 2754	. . . . . . . . 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 631	. . . . . . . 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 341	. . . . . . 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 7275	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘))
41	40	iuneq1d 4951	. . . . . . . . . 10 ⊢ (𝑧 = 𝑘 → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
42	41	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑧 = 𝑘 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol))
43	41	ineq2d 4146	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑘 → (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
44	43	fveq2d 6770	. . . . . . . . . 10 ⊢ (𝑧 = 𝑘 → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
45		fveq2 6766	. . . . . . . . . 10 ⊢ (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘))
46	44, 45	eqeq12d 2754	. . . . . . . . 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 631	. . . . . . . 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 341	. . . . . . 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 7275	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1)))
50	49	iuneq1d 4951	. . . . . . . . . 10 ⊢ (𝑧 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
51	50	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑧 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol))
52	50	ineq2d 4146	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑘 + 1) → (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)))
53	52	fveq2d 6770	. . . . . . . . . 10 ⊢ (𝑧 = (𝑘 + 1) → (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))))
54		fveq2 6766	. . . . . . . . . 10 ⊢ (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1)))
55	53, 54	eqeq12d 2754	. . . . . . . . 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 631	. . . . . . . 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 341	. . . . . . 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 12360	. . . . . . . . . . 11 ⊢ 1 ∈ ℤ
59		fzsn 13308	. . . . . . . . . . 11 ⊢ (1 ∈ ℤ → (1...1) = {1})
60		iuneq1 4940	. . . . . . . . . . 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 10981	. . . . . . . . . . 11 ⊢ 1 ∈ V
63		fveq2 6766	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1))
64	62, 63	iunxsn 5019	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1)
65	61, 64	eqtri 2766	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1)
66		1nn 11994	. . . . . . . . . 10 ⊢ 1 ∈ ℕ
67		ffvelrn 6951	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶dom vol ∧ 1 ∈ ℕ) → (𝐹‘1) ∈ dom vol)
68	14, 66, 67	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘1) ∈ dom vol)
69	65, 68	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol)
70	63	ineq2d 4146	. . . . . . . . . . . 12 ⊢ (𝑛 = 1 → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1)))
71	70	fveq2d 6770	. . . . . . . . . . 11 ⊢ (𝑛 = 1 → (vol‘(𝐸 ∩ (𝐹‘𝑛))) = (vol‘(𝐸 ∩ (𝐹‘1))))
72		voliunlem1.6	. . . . . . . . . . 11 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹‘𝑛))))
73		fvex 6779	. . . . . . . . . . 11 ⊢ (vol*‘(𝐸 ∩ (𝐹‘1))) ∈ V
74	71, 72, 73	fvmpt 6867	. . . . . . . . . 10 ⊢ (1 ∈ ℕ → (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))))
75	66, 74	ax-mp 5	. . . . . . . . 9 ⊢ (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1)))
76		seq1 13744	. . . . . . . . . 10 ⊢ (1 ∈ ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1))
77	58, 76	ax-mp 5	. . . . . . . . 9 ⊢ (seq1( + , 𝐻)‘1) = (𝐻‘1)
78	65	ineq2i 4143	. . . . . . . . . 10 ⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1))
79	78	fveq2i 6769	. . . . . . . . 9 ⊢ (vol‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (vol‘(𝐸 ∩ (𝐹‘1)))
80	75, 77, 79	3eqtr4ri 2777	. . . . . . . 8 ⊢ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)
81	69, 80	jctir 521	. . . . . . 7 ⊢ (𝜑 → (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (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)
84	14, 82, 83	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol)
85		unmbl 24711	. . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
89		nnuz 12631	. . . . . . . . . . . . . . 15 ⊢ ℕ = (ℤ_≥‘1)
90	88, 89	eleqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 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))(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 5022	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛))
95		ovex 7300	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 + 1) ∈ V
96		fveq2 6766	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)))
97	95, 96	iunxsn 5019	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))
98	97	uneq2i 4093	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
99	94, 98	eqtri 2766	. . . . . . . . . . . . 13 ⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))
100	93, 99	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))))
101	100	eleq1d 2823	. . . . . . . . . . 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 258	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol))
103		oveq1 7274	. . . . . . . . . . 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 4162	. . . . . . . . . . . . . . 15 ⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸
105	104, 26	sstrid 3931	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ)
106		ovolsscl 24660	. . . . . . . . . . . . . . 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 1467	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ)
108		mblsplit 24706	. . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
111		inss2 4163	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1))
112	82	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ)
113	112, 89	eleqtrdi 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ (ℤ_≥‘1))
114		eluzfz2 13274	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑘 + 1) ∈ (ℤ_≥‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1)))
115	96	ssiun2s 4977	. . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))
118		df-ss 3903	. . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
120	110, 119	eqtrid 2790	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
121	120	fveq2d 6770	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
122		indif2 4204	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∩ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))
123		uncom 4086	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
124	100, 123	eqtr2di 2795	. . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖))
127	112	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ)
128	13	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ)
129	128	nnred 11998	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ)
130		elfzle2 13270	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ≤ 𝑘)
131	130	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ≤ 𝑘)
132	88	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ)
133		nnleltp1 12385	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1)))
134	128, 132, 133	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1)))
135	131, 134	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1))
136	129, 135	gtned 11120	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛)
137		fveq2 6766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = (𝑘 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑘 + 1)))
138		fveq2 6766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛))
139	137, 138	disji2 5055	. . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅)
141	140	iuneq2dv 4948	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∪ 𝑛 ∈ (1...𝑘)∅)
142		iunin2 4999	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
143		iun0 4990	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑛 ∈ (1...𝑘)∅ = ∅
144	141, 142, 143	3eqtr3g 2801	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅)
145		uneqdifeq 4423	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))
148	147	ineq2d 4146	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
149	122, 148	eqtr3id 2792	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))
150	149	fveq2d 6770	. . . . . . . . . . . . . 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 7285	. . . . . . . . . . . . 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 4162	. . . . . . . . . . . . . . . 16 ⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸
153		ovolsscl 24660	. . . . . . . . . . . . . . . 16 ⊢ (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol‘𝐸) ∈ ℝ) → (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
154	152, 2, 4, 153	mp3an2ani 1467	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ)
155	154	recnd 11013	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ)
156	12	recnd 11013	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℂ)
157	155, 156	addcomd 11187	. . . . . . . . . . . . 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 2782	. . . . . . . . . . . 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 13746	. . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1))))
162	161	fveq2d 6770	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = (𝑘 + 1) → (vol‘(𝐸 ∩ (𝐹‘𝑛))) = (vol‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))
163		fvex 6779	. . . . . . . . . . . . . . . 16 ⊢ (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ V
164	162, 72, 163	fvmpt 6867	. . . . . . . . . . . . . . 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 7283	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
167	160, 166	eqtrd 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))
168	158, 167	eqeq12d 2754	. . . . . . . . . . 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	syl5ibr 245	. . . . . . . . . 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 609	. . . . . . . . 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 414	. . . . . . . 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 12001	. . . . . 6 ⊢ (𝑘 ∈ ℕ → (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))))
174	173	impcom 408	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))
175	174	simprd 496	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))
176	175	eqcomd 2744	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))
177	176	oveq1d 7282	. 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 495	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol)
179		mblsplit 24706	. . 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 1370	. 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 5105	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 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 ℝcr 10880 1c1 10882 + caddc 10884 < clt 11019 ≤ cle 11020 ℕcn 11983 ℤcz 12329 ℤ_≥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

W3C validator