Theorem volfiniun 24616

Description: The volume of a disjoint finite union of measurable sets is the sum of the measures. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Assertion

Ref	Expression
volfiniun	⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵) → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))

Distinct variable group:   𝐴,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem volfiniun

Dummy variables 𝑚 𝑛 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		raleq 3333	. . . . 5 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2		disjeq1 5042	. . . . 5 ⊢ (𝑤 = ∅ → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ ∅ 𝐵))
3	1, 2	anbi12d 630	. . . 4 ⊢ (𝑤 = ∅ → ((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) ↔ (∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵)))
4		iuneq1 4937	. . . . . 6 ⊢ (𝑤 = ∅ → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
5	4	fveq2d 6760	. . . . 5 ⊢ (𝑤 = ∅ → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ ∅ 𝐵))
6		sumeq1 15328	. . . . 5 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
7	5, 6	eqeq12d 2754	. . . 4 ⊢ (𝑤 = ∅ → ((vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵) ↔ (vol‘∪ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)))
8	3, 7	imbi12d 344	. . 3 ⊢ (𝑤 = ∅ → (((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵)) ↔ ((∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵) → (vol‘∪ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))))
9		raleq 3333	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10		disjeq1 5042	. . . . 5 ⊢ (𝑤 = 𝑦 → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ 𝑦 𝐵))
11	9, 10	anbi12d 630	. . . 4 ⊢ (𝑤 = 𝑦 → ((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) ↔ (∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑦 𝐵)))
12		iuneq1 4937	. . . . . 6 ⊢ (𝑤 = 𝑦 → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
13	12	fveq2d 6760	. . . . 5 ⊢ (𝑤 = 𝑦 → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ 𝑦 𝐵))
14		sumeq1 15328	. . . . 5 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵))
15	13, 14	eqeq12d 2754	. . . 4 ⊢ (𝑤 = 𝑦 → ((vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵) ↔ (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵)))
16	11, 15	imbi12d 344	. . 3 ⊢ (𝑤 = 𝑦 → (((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵)) ↔ ((∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑦 𝐵) → (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵))))
17		raleq 3333	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18		disjeq1 5042	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
19	17, 18	anbi12d 630	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) ↔ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
20		iuneq1 4937	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
21	20	fveq2d 6760	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22		sumeq1 15328	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
23	21, 22	eqeq12d 2754	. . . 4 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵) ↔ (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
24	19, 23	imbi12d 344	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵)) ↔ ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
25		raleq 3333	. . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26		disjeq1 5042	. . . . 5 ⊢ (𝑤 = 𝐴 → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ 𝐴 𝐵))
27	25, 26	anbi12d 630	. . . 4 ⊢ (𝑤 = 𝐴 → ((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) ↔ (∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵)))
28		iuneq1 4937	. . . . . 6 ⊢ (𝑤 = 𝐴 → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
29	28	fveq2d 6760	. . . . 5 ⊢ (𝑤 = 𝐴 → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ 𝐴 𝐵))
30		sumeq1 15328	. . . . 5 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))
31	29, 30	eqeq12d 2754	. . . 4 ⊢ (𝑤 = 𝐴 → ((vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵) ↔ (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)))
32	27, 31	imbi12d 344	. . 3 ⊢ (𝑤 = 𝐴 → (((∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑤 𝐵) → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = Σ𝑘 ∈ 𝑤 (vol‘𝐵)) ↔ ((∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵) → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))))
33		0mbl 24608	. . . . . . 7 ⊢ ∅ ∈ dom vol
34		mblvol 24599	. . . . . . 7 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
35	33, 34	ax-mp 5	. . . . . 6 ⊢ (vol‘∅) = (vol*‘∅)
36		ovol0 24562	. . . . . 6 ⊢ (vol*‘∅) = 0
37	35, 36	eqtri 2766	. . . . 5 ⊢ (vol‘∅) = 0
38		0iun 4988	. . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
39	38	fveq2i 6759	. . . . 5 ⊢ (vol‘∪ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40		sum0 15361	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
41	37, 39, 40	3eqtr4i 2776	. . . 4 ⊢ (vol‘∪ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)
42	41	a1i 11	. . 3 ⊢ ((∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵) → (vol‘∪ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
43		ssun1 4102	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
44		ssralv 3983	. . . . . . 7 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → ∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
45	43, 44	ax-mp 5	. . . . . 6 ⊢ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → ∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ))
46		disjss1 5041	. . . . . . 7 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 → Disj 𝑘 ∈ 𝑦 𝐵))
47	43, 46	ax-mp 5	. . . . . 6 ⊢ (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 → Disj 𝑘 ∈ 𝑦 𝐵)
48	45, 47	anim12i 612	. . . . 5 ⊢ ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑦 𝐵))
49	48	imim1i 63	. . . 4 ⊢ (((∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑦 𝐵) → (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵)) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵)))
50		oveq1 7262	. . . . . . . 8 ⊢ ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) = Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) → ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)) = (Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)))
51		iunxun 5019	. . . . . . . . . . . 12 ⊢ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
52		vex 3426	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
53		csbeq1 3831	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
54	52, 53	iunxsn 5016	. . . . . . . . . . . . 13 ⊢ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
55	54	uneq2i 4090	. . . . . . . . . . . 12 ⊢ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
56	51, 55	eqtri 2766	. . . . . . . . . . 11 ⊢ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
57	56	fveq2i 6759	. . . . . . . . . 10 ⊢ (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
58		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚𝐵
59		nfcsb1v 3853	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
60		csbeq1a 3842	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
61	58, 59, 60	cbviun 4962	. . . . . . . . . . . 12 ⊢ ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
62		simpll 763	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑦 ∈ Fin)
63		simprl 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ))
64		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → 𝐵 ∈ dom vol)
65	64	ralimi 3086	. . . . . . . . . . . . . . 15 ⊢ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol)
66	63, 65	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol)
67		ssralv 3983	. . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol → ∀𝑘 ∈ 𝑦 𝐵 ∈ dom vol))
68	43, 66, 67	mpsyl 68	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘 ∈ 𝑦 𝐵 ∈ dom vol)
69		finiunmbl 24613	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Fin ∧ ∀𝑘 ∈ 𝑦 𝐵 ∈ dom vol) → ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol)
70	62, 68, 69	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∪ 𝑘 ∈ 𝑦 𝐵 ∈ dom vol)
71	61, 70	eqeltrrid 2844	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol)
72		ssun2 4103	. . . . . . . . . . . . . 14 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
73		vsnid 4595	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ {𝑧}
74	72, 73	sselii 3914	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
75		nfcsb1v 3853	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
76	75	nfel1 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol
77		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘vol
78	77, 75	nffv 6766	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(vol‘⦋𝑧 / 𝑘⦌𝐵)
79	78	nfel1 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
80	76, 79	nfan 1903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81		csbeq1a 3842	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
82	81	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol))
83	81	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘⦋𝑧 / 𝑘⦌𝐵))
84	83	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
85	82, 84	anbi12d 630	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
86	80, 85	rspc 3539	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → (⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
87	74, 63, 86	mpsyl 68	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
88	87	simpld 494	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol)
89		simplr 765	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑧 ∈ 𝑦)
90		elin 3899	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∩ ⦋𝑧 / 𝑘⦌𝐵) ↔ (𝑤 ∈ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵))
91		eliun 4925	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ↔ ∃𝑚 ∈ 𝑦 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵)
92		simplrr 774	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
93		nfcv 2906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛𝐵
94		nfcsb1v 3853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
95		csbeq1a 3842	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
96	93, 94, 95	cbvdisj 5045	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ Disj 𝑛 ∈ (𝑦 ∪ {𝑧})⦋𝑛 / 𝑘⦌𝐵)
97	92, 96	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → Disj 𝑛 ∈ (𝑦 ∪ {𝑧})⦋𝑛 / 𝑘⦌𝐵)
98		simpr1 1192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑚 ∈ 𝑦)
99		elun1 4106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
100	98, 99	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑚 ∈ (𝑦 ∪ {𝑧}))
101	74	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
102		simpr2 1193	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵)
103		simpr3 1194	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)
104		csbeq1 3831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
105		csbeq1 3831	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑧 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
106	104, 105	disji 5053	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Disj 𝑛 ∈ (𝑦 ∪ {𝑧})⦋𝑛 / 𝑘⦌𝐵 ∧ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) ∧ (𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑚 = 𝑧)
107	97, 100, 101, 102, 103, 106	syl122anc 1377	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑚 = 𝑧)
108	107, 98	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚 ∈ 𝑦 ∧ 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵)) → 𝑧 ∈ 𝑦)
109	108	3exp2 1352	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑚 ∈ 𝑦 → (𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 → (𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵 → 𝑧 ∈ 𝑦))))
110	109	rexlimdv 3211	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (∃𝑚 ∈ 𝑦 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵 → (𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵 → 𝑧 ∈ 𝑦)))
111	91, 110	syl5bi 241	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑤 ∈ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 → (𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵 → 𝑧 ∈ 𝑦)))
112	111	impd 410	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((𝑤 ∈ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵) → 𝑧 ∈ 𝑦))
113	90, 112	syl5bi 241	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑤 ∈ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∩ ⦋𝑧 / 𝑘⦌𝐵) → 𝑧 ∈ 𝑦))
114	89, 113	mtod 197	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑤 ∈ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∩ ⦋𝑧 / 𝑘⦌𝐵))
115	114	eq0rdv 4335	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∩ ⦋𝑧 / 𝑘⦌𝐵) = ∅)
116		mblvol 24599	. . . . . . . . . . . . 13 ⊢ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol → (vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵))
117	71, 116	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵))
118		nfv 1918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑚(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)
119	59	nfel1 2922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol
120	77, 59	nffv 6766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(vol‘⦋𝑚 / 𝑘⦌𝐵)
121	120	nfel1 2922	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘(vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
122	119, 121	nfan 1903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
123	60	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ ⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol))
124	60	fveq2d 6760	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘⦋𝑚 / 𝑘⦌𝐵))
125	124	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
126	123, 125	anbi12d 630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
127	118, 122, 126	cbvralw 3363	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
128	63, 127	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})(⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
129	128	r19.21bi 3132	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
130	129	simpld 494	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → ⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol)
131		mblss 24600	. . . . . . . . . . . . . . . . 17 ⊢ (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol → ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
132	130, 131	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
133	99, 132	sylan2 592	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ 𝑦) → ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
134	133	ralrimiva 3107	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
135		iunss 4971	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
136	134, 135	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
137		mblvol 24599	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol → (vol‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
138	137	eleq1d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol → ((vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
139	138	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ) → (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
140	129, 139	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
141	99, 140	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ 𝑦) → (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
142	62, 141	fsumrecl 15374	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
143	131	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ) → ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
144	143, 139	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
145	144	ralimi 3086	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑚 ∈ (𝑦 ∪ {𝑧})(⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
146	128, 145	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
147		ssralv 3983	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ) → ∀𝑚 ∈ 𝑦 (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
148	43, 146, 147	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚 ∈ 𝑦 (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
149		ovolfiniun 24570	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Fin ∧ ∀𝑚 ∈ 𝑦 (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)) → (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ≤ Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵))
150	62, 148, 149	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ≤ Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵))
151		ovollecl 24552	. . . . . . . . . . . . 13 ⊢ ((∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ ∧ (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ≤ Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵)) → (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
152	136, 142, 150, 151	syl3anc 1369	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
153	117, 152	eqeltrd 2839	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
154	87	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
155		volun 24614	. . . . . . . . . . 11 ⊢ (((∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ ⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∩ ⦋𝑧 / 𝑘⦌𝐵) = ∅) ∧ ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)) → (vol‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)) = ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)))
156	71, 88, 115, 153, 154, 155	syl32anc 1376	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)) = ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)))
157	57, 156	syl5eq 2791	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)))
158		disjsn 4644	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
159	89, 158	sylibr 233	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
160		eqidd 2739	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
161		snfi 8788	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
162		unfi 8917	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
163	62, 161, 162	sylancl 585	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
164	129	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
165	164	recnd 10934	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℂ)
166	159, 160, 163, 165	fsumsplit 15381	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵) = (Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) + Σ𝑚 ∈ {𝑧} (vol‘⦋𝑚 / 𝑘⦌𝐵)))
167	154	recnd 10934	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
168	53	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑧 → (vol‘⦋𝑚 / 𝑘⦌𝐵) = (vol‘⦋𝑧 / 𝑘⦌𝐵))
169	168	sumsn 15386	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol‘⦋𝑚 / 𝑘⦌𝐵) = (vol‘⦋𝑧 / 𝑘⦌𝐵))
170	52, 167, 169	sylancr 586	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ {𝑧} (vol‘⦋𝑚 / 𝑘⦌𝐵) = (vol‘⦋𝑧 / 𝑘⦌𝐵))
171	170	oveq2d 7271	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) + Σ𝑚 ∈ {𝑧} (vol‘⦋𝑚 / 𝑘⦌𝐵)) = (Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)))
172	166, 171	eqtrd 2778	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵) = (Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)))
173	157, 172	eqeq12d 2754	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵) ↔ ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵)) = (Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) + (vol‘⦋𝑧 / 𝑘⦌𝐵))))
174	50, 173	syl5ibr 245	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) = Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵) → (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵)))
175	61	fveq2i 6759	. . . . . . . 8 ⊢ (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
176		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑚(vol‘𝐵)
177	176, 120, 124	cbvsumi 15337	. . . . . . . 8 ⊢ Σ𝑘 ∈ 𝑦 (vol‘𝐵) = Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵)
178	175, 177	eqeq12i 2756	. . . . . . 7 ⊢ ((vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵) ↔ (vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) = Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵))
179	58, 59, 60	cbviun 4962	. . . . . . . . 9 ⊢ ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
180	179	fveq2i 6759	. . . . . . . 8 ⊢ (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
181	176, 120, 124	cbvsumi 15337	. . . . . . . 8 ⊢ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵)
182	180, 181	eqeq12i 2756	. . . . . . 7 ⊢ ((vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) ↔ (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵))
183	174, 178, 182	3imtr4g 295	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵) → (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
184	183	ex 412	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → ((vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵) → (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
185	184	a2d 29	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵)) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
186	49, 185	syl5 34	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝑦 𝐵) → (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵)) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
187	8, 16, 24, 32, 42, 186	findcard2s 8910	. 2 ⊢ (𝐴 ∈ Fin → ((∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵) → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵)))
188	187	3impib 1114	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ 𝐴 𝐵) → (vol‘∪ 𝑘 ∈ 𝐴 𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422  ⦋csb 3828   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  {csn 4558  ∪ ciun 4921  Disj wdisj 5035   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  ℂcc 10800  ℝcr 10801  0cc0 10802   + caddc 10805   ≤ cle 10941  Σcsu 15325  vol*covol 24531  volcvol 24532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-disj 5036  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-sup 9131  df-inf 9132  df-oi 9199  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-z 12250  df-uz 12512  df-q 12618  df-rp 12660  df-xadd 12778  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-sum 15326  df-xmet 20503  df-met 20504  df-ovol 24533  df-vol 24534

This theorem is referenced by:  uniioovol  24648  uniioombllem4  24655  itg1addlem1  24761  volfiniune  32098