Theorem volfiniun 25601

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 3331	. . . . 5 ⊢ (𝑤 = ∅ → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2		disjeq1 5140	. . . . 5 ⊢ (𝑤 = ∅ → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ ∅ 𝐵))
3	1, 2	anbi12d 631	. . . 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 5031	. . . . . 6 ⊢ (𝑤 = ∅ → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
5	4	fveq2d 6924	. . . . 5 ⊢ (𝑤 = ∅ → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ ∅ 𝐵))
6		sumeq1 15737	. . . . 5 ⊢ (𝑤 = ∅ → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
7	5, 6	eqeq12d 2756	. . . 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 3331	. . . . 5 ⊢ (𝑤 = 𝑦 → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10		disjeq1 5140	. . . . 5 ⊢ (𝑤 = 𝑦 → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ 𝑦 𝐵))
11	9, 10	anbi12d 631	. . . 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 5031	. . . . . 6 ⊢ (𝑤 = 𝑦 → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
13	12	fveq2d 6924	. . . . 5 ⊢ (𝑤 = 𝑦 → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ 𝑦 𝐵))
14		sumeq1 15737	. . . . 5 ⊢ (𝑤 = 𝑦 → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ 𝑦 (vol‘𝐵))
15	13, 14	eqeq12d 2756	. . . 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 3331	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18		disjeq1 5140	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
19	17, 18	anbi12d 631	. . . 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 5031	. . . . . 6 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
21	20	fveq2d 6924	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22		sumeq1 15737	. . . . 5 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
23	21, 22	eqeq12d 2756	. . . 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 3331	. . . . 5 ⊢ (𝑤 = 𝐴 → (∀𝑘 ∈ 𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26		disjeq1 5140	. . . . 5 ⊢ (𝑤 = 𝐴 → (Disj 𝑘 ∈ 𝑤 𝐵 ↔ Disj 𝑘 ∈ 𝐴 𝐵))
27	25, 26	anbi12d 631	. . . 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 5031	. . . . . 6 ⊢ (𝑤 = 𝐴 → ∪ 𝑘 ∈ 𝑤 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
29	28	fveq2d 6924	. . . . 5 ⊢ (𝑤 = 𝐴 → (vol‘∪ 𝑘 ∈ 𝑤 𝐵) = (vol‘∪ 𝑘 ∈ 𝐴 𝐵))
30		sumeq1 15737	. . . . 5 ⊢ (𝑤 = 𝐴 → Σ𝑘 ∈ 𝑤 (vol‘𝐵) = Σ𝑘 ∈ 𝐴 (vol‘𝐵))
31	29, 30	eqeq12d 2756	. . . 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 25593	. . . . . . 7 ⊢ ∅ ∈ dom vol
34		mblvol 25584	. . . . . . 7 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
35	33, 34	ax-mp 5	. . . . . 6 ⊢ (vol‘∅) = (vol*‘∅)
36		ovol0 25547	. . . . . 6 ⊢ (vol*‘∅) = 0
37	35, 36	eqtri 2768	. . . . 5 ⊢ (vol‘∅) = 0
38		0iun 5086	. . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
39	38	fveq2i 6923	. . . . 5 ⊢ (vol‘∪ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40		sum0 15769	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
41	37, 39, 40	3eqtr4i 2778	. . . 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 4201	. . . . . . 7 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
44		ssralv 4077	. . . . . . 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 5139	. . . . . . 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 7455	. . . . . . . 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 5117	. . . . . . . . . . . 12 ⊢ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
52		vex 3492	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
53		csbeq1 3924	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
54	52, 53	iunxsn 5114	. . . . . . . . . . . . 13 ⊢ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
55	54	uneq2i 4188	. . . . . . . . . . . 12 ⊢ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
56	51, 55	eqtri 2768	. . . . . . . . . . 11 ⊢ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
57	56	fveq2i 6923	. . . . . . . . . 10 ⊢ (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
58		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚𝐵
59		nfcsb1v 3946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
60		csbeq1a 3935	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
61	58, 59, 60	cbviun 5059	. . . . . . . . . . . 12 ⊢ ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
62		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑦 ∈ Fin)
63		simprl 770	. . . . . . . . . . . . . . 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 3089	. . . . . . . . . . . . . . 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 4077	. . . . . . . . . . . . . 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 25598	. . . . . . . . . . . . 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 2849	. . . . . . . . . . 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 4202	. . . . . . . . . . . . . 14 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
73		vsnid 4685	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ {𝑧}
74	72, 73	sselii 4005	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
75		nfcsb1v 3946	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
76	75	nfel1 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol
77		nfcv 2908	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘vol
78	77, 75	nffv 6930	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(vol‘⦋𝑧 / 𝑘⦌𝐵)
79	78	nfel1 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
80	76, 79	nfan 1898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81		csbeq1a 3935	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
82	81	eleq1d 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ ⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol))
83	81	fveq2d 6924	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘⦋𝑧 / 𝑘⦌𝐵))
84	83	eleq1d 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
85	82, 84	anbi12d 631	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (⦋𝑧 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
86	80, 85	rspc 3623	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑧 ∈ 𝑦)
90		elin 3992	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∩ ⦋𝑧 / 𝑘⦌𝐵) ↔ (𝑤 ∈ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∧ 𝑤 ∈ ⦋𝑧 / 𝑘⦌𝐵))
91		eliun 5019	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ↔ ∃𝑚 ∈ 𝑦 𝑤 ∈ ⦋𝑚 / 𝑘⦌𝐵)
92		simplrr 777	. . . . . . . . . . . . . . . . . . . . 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 2908	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑛𝐵
94		nfcsb1v 3946	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
95		csbeq1a 3935	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
96	93, 94, 95	cbvdisj 5143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ Disj 𝑛 ∈ (𝑦 ∪ {𝑧})⦋𝑛 / 𝑘⦌𝐵)
97	92, 96	sylib 218	. . . . . . . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . . . . . . . . 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 4205	. . . . . . . . . . . . . . . . . . . . 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 1195	. . . . . . . . . . . . . . . . . . . 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 1196	. . . . . . . . . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑚 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
105		csbeq1 3924	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑧 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
106	104, 105	disji 5151	. . . . . . . . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . . . 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 198	. . . . . . . . . . . 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 4430	. . . . . . . . . . 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 25584	. . . . . . . . . . . . 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 1913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑚(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)
119	59	nfel1 2925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol
120	77, 59	nffv 6930	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑘(vol‘⦋𝑚 / 𝑘⦌𝐵)
121	120	nfel1 2925	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑘(vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
122	119, 121	nfan 1898	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
123	60	eleq1d 2829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ ⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol))
124	60	fveq2d 6924	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘⦋𝑚 / 𝑘⦌𝐵))
125	124	eleq1d 2829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
126	123, 125	anbi12d 631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol ∧ (vol‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
127	118, 122, 126	cbvralw 3312	. . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . 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 3257	. . . . . . . . . . . . . . . . . 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 25585	. . . . . . . . . . . . . . . . 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 3152	. . . . . . . . . . . . . 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 5068	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
136	134, 135	sylibr 234	. . . . . . . . . . . . 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 25584	. . . . . . . . . . . . . . . . . 18 ⊢ (⦋𝑚 / 𝑘⦌𝐵 ∈ dom vol → (vol‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
138	137	eleq1d 2829	. . . . . . . . . . . . . . . . 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 15782	. . . . . . . . . . . . 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 3089	. . . . . . . . . . . . . . . 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 4077	. . . . . . . . . . . . . . 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 25555	. . . . . . . . . . . . . 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 25537	. . . . . . . . . . . . 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 1371	. . . . . . . . . . . 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 2844	. . . . . . . . . . 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 25599	. . . . . . . . . . 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 1378	. . . . . . . . . 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	eqtrid 2792	. . . . . . . . 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 4736	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
159	89, 158	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
160		eqidd 2741	. . . . . . . . . . 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 9109	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
162		unfi 9238	. . . . . . . . . . . 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 11318	. . . . . . . . . . 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 15789	. . . . . . . . . 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 11318	. . . . . . . . . . . 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 6924	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑧 → (vol‘⦋𝑚 / 𝑘⦌𝐵) = (vol‘⦋𝑧 / 𝑘⦌𝐵))
169	168	sumsn 15794	. . . . . . . . . . . 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 7464	. . . . . . . . . 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 2780	. . . . . . . . 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 2756	. . . . . . . 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	imbitrrid 246	. . . . . . 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 6923	. . . . . . . 8 ⊢ (vol‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
176		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑚(vol‘𝐵)
177	124, 176, 120	cbvsum 15743	. . . . . . . 8 ⊢ Σ𝑘 ∈ 𝑦 (vol‘𝐵) = Σ𝑚 ∈ 𝑦 (vol‘⦋𝑚 / 𝑘⦌𝐵)
178	175, 177	eqeq12i 2758	. . . . . . 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 5059	. . . . . . . . 9 ⊢ ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
180	179	fveq2i 6923	. . . . . . . 8 ⊢ (vol‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
181	124, 176, 120	cbvsum 15743	. . . . . . . 8 ⊢ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘⦋𝑚 / 𝑘⦌𝐵)
182	180, 181	eqeq12i 2758	. . . . . . 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 296	. . . . . 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 9231	. 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 1116	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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488  ⦋csb 3921   ∪ cun 3974   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∪ ciun 5015  Disj wdisj 5133   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  Fincfn 9003  ℂcc 11182  ℝcr 11183  0cc0 11184   + caddc 11187   ≤ cle 11325  Σcsu 15734  vol*covol 25516  volcvol 25517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261  ax-pre-sup 11262

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-sup 9511  df-inf 9512  df-oi 9579  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-z 12640  df-uz 12904  df-q 13014  df-rp 13058  df-xadd 13176  df-ioo 13411  df-ico 13413  df-icc 13414  df-fz 13568  df-fzo 13712  df-fl 13843  df-seq 14053  df-exp 14113  df-hash 14380  df-cj 15148  df-re 15149  df-im 15150  df-sqrt 15284  df-abs 15285  df-clim 15534  df-sum 15735  df-xmet 21380  df-met 21381  df-ovol 25518  df-vol 25519

This theorem is referenced by:  uniioovol  25633  uniioombllem4  25640  itg1addlem1  25746  volfiniune  34194