Theorem ovolfiniun 25802

Description: The Lebesgue outer measure function is finitely sub-additive. Finite sum version. (Contributed by Mario Carneiro, 19-Jun-2014.)

Assertion

Ref	Expression
ovolfiniun	⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵))

Distinct variable group:   𝐴,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem ovolfiniun

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

Step	Hyp	Ref	Expression
1		raleq 3317	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2		iuneq1 4968	. . . . . 6 ⊢ (𝑥 = ∅ → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
3	2	fveq2d 6881	. . . . 5 ⊢ (𝑥 = ∅ → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ ∅ 𝐵))
4		sumeq1 15836	. . . . 5 ⊢ (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
5	3, 4	breq12d 5116	. . . 4 ⊢ (𝑥 = ∅ → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
6	1, 5	imbi12d 347	. . 3 ⊢ (𝑥 = ∅ → ((∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵)) ↔ (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))))
7		raleq 3317	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8		iuneq1 4968	. . . . . 6 ⊢ (𝑥 = 𝑦 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
9	8	fveq2d 6881	. . . . 5 ⊢ (𝑥 = 𝑦 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝑦 𝐵))
10		sumeq1 15836	. . . . 5 ⊢ (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝑦 (vol*‘𝐵))
11	9, 10	breq12d 5116	. . . 4 ⊢ (𝑥 = 𝑦 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)))
12	7, 11	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑦 → ((∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵)) ↔ (∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))))
13		raleq 3317	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14		iuneq1 4968	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
15	14	fveq2d 6881	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16		sumeq1 15836	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
17	15, 16	breq12d 5116	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
18	13, 17	imbi12d 347	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵)) ↔ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))))
19		raleq 3317	. . . 4 ⊢ (𝑥 = 𝐴 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20		iuneq1 4968	. . . . . 6 ⊢ (𝑥 = 𝐴 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
21	20	fveq2d 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝐴 𝐵))
22		sumeq1 15836	. . . . 5 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
23	21, 22	breq12d 5116	. . . 4 ⊢ (𝑥 = 𝐴 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
24	19, 23	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵)) ↔ (∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵))))
25		0le0 12425	. . . . 5 ⊢ 0 ≤ 0
26		0iun 5021	. . . . . . 7 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅
27	26	fveq2i 6880	. . . . . 6 ⊢ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28		ovol0 25794	. . . . . 6 ⊢ (vol*‘∅) = 0
29	27, 28	eqtri 2784	. . . . 5 ⊢ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = 0
30		sum0 15867	. . . . 5 ⊢ Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
31	25, 29, 30	3brtr4i 5135	. . . 4 ⊢ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
32	31	a1i 11	. . 3 ⊢ (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33		ssun1 4124	. . . . . 6 ⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧})
34		ssralv 4000	. . . . . 6 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
35	33, 34	ax-mp 5	. . . . 5 ⊢ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
36	35	imim1i 64	. . . 4 ⊢ ((∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)))
37		simprl 783	. . . . . . . . . . . . 13 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
38		nfcsb1v 3871	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
39		nfcv 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘ℝ
40	38, 39	nfss 3924	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ
41		nfcv 2923	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘vol*
42	41, 38	nffv 6887	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵)
43	42	nfel1 2939	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
44	40, 43	nfan 1932	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
45		csbeq1a 3861	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
46	45	sseq1d 3962	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ))
47	45	fveq2d 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
48	47	eleq1d 2846	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
49	46, 48	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
50	44, 49	rspc 3565	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
51	37, 50	mpan9 516	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
52	51	simpld 500	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
53	52	ralrimiva 3155	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
54		iunss 5003	. . . . . . . . . 10 ⊢ (∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
55	53, 54	sylibr 237	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
56		iunss1 4966	. . . . . . . . . . . . 13 ⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
57	33, 56	ax-mp 5	. . . . . . . . . . . 12 ⊢ ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
58	57, 55	sstrid 3942	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
59		simpll 779	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → 𝑦 ∈ Fin)
60		elun1 4128	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
61	51	simprd 501	. . . . . . . . . . . . 13 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
62	60, 61	sylan2 605	. . . . . . . . . . . 12 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ 𝑦) → (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
63	59, 62	fsumrecl 15880	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
64		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))
65		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚𝐵
66	65, 38, 45	cbviun 4993	. . . . . . . . . . . . 13 ⊢ ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
67	66	fveq2i 6880	. . . . . . . . . . . 12 ⊢ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
68		nfcv 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(vol*‘𝐵)
69	47, 68, 42	cbvsum 15842	. . . . . . . . . . . 12 ⊢ Σ𝑘 ∈ 𝑦 (vol*‘𝐵) = Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵)
70	64, 67, 69	3brtr3g 5138	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ≤ Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵))
71		ovollecl 25784	. . . . . . . . . . 11 ⊢ ((∪ 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)
72	58, 63, 70, 71	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
73		ssun2 4125	. . . . . . . . . . . . 13 ⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧})
74		vsnid 4624	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ {𝑧}
75	73, 74	sselii 3928	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ (𝑦 ∪ {𝑧})
76		nfcsb1v 3871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
77	76, 39	nfss 3924	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ
78	41, 76	nffv 6887	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵)
79	78	nfel1 2939	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
80	77, 79	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81		csbeq1a 3861	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
82	81	sseq1d 3962	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ))
83	81	fveq2d 6881	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
84	83	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
85	82, 84	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
86	80, 85	rspc 3565	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
87	75, 37, 86	mpsyl 69	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
88	87	simprd 501	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
89	72, 88	readdcld 11319	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)) ∈ ℝ)
90		iunxun 5054	. . . . . . . . . . . 12 ⊢ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
91		vex 3455	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
92		csbeq1 3850	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
93	91, 92	iunxsn 5051	. . . . . . . . . . . . 13 ⊢ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
94	93	uneq2i 4112	. . . . . . . . . . . 12 ⊢ (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
95	90, 94	eqtri 2784	. . . . . . . . . . 11 ⊢ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
96	95	fveq2i 6880	. . . . . . . . . 10 ⊢ (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol*‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
97		ovolun 25800	. . . . . . . . . . 11 ⊢ (((∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ) ∧ (⦋𝑧 / 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)))
98	58, 72, 87, 97	syl21anc 851	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)) ≤ ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)))
99	96, 98	eqbrtrid 5140	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) ≤ ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)))
100		ovollecl 25784	. . . . . . . . 9 ⊢ ((∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)) ∈ ℝ ∧ (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) ≤ ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵))) → (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
101	55, 89, 99, 100	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
102		snfi 9055	. . . . . . . . . . 11 ⊢ {𝑧} ∈ Fin
103		unfi 9170	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104	102, 103	mpan2 704	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105	104	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) ∈ Fin)
106	105, 61	fsumrecl 15880	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
107	72, 63, 88, 70	leadd1dd 11911	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)) ≤ (Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)))
108		simplr 781	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ¬ 𝑧 ∈ 𝑦)
109		disjsn 4672	. . . . . . . . . . . 12 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
110	108, 109	sylibr 237	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (𝑦 ∩ {𝑧}) = ∅)
111		eqidd 2762	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
112	61	recnd 11318	. . . . . . . . . . 11 ⊢ ((((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℂ)
113	110, 111, 105, 112	fsumsplit 15887	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵) = (Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) + Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵)))
114	88	recnd 11318	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ)
115	92	fveq2d 6881	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑧 → (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
116	115	sumsn 15892	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
117	91, 114, 116	sylancr 599	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
118	117	oveq2d 7428	. . . . . . . . . 10 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) + Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵)) = (Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)))
119	113, 118	eqtrd 2796	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵) = (Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)))
120	107, 119	breqtrrd 5133	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → ((vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵) + (vol*‘⦋𝑧 / 𝑘⦌𝐵)) ≤ Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵))
121	101, 89, 106, 99, 120	letrd 11448	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) ≤ Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵))
122	65, 38, 45	cbviun 4993	. . . . . . . 8 ⊢ ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
123	122	fveq2i 6880	. . . . . . 7 ⊢ (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
124	47, 68, 42	cbvsum 15842	. . . . . . 7 ⊢ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵)
125	121, 123, 124	3brtr4g 5139	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
126	125	exp32 426	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵) → (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))))
127	126	a2d 30	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))))
128	36, 127	syl5 35	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))))
129	6, 12, 18, 24, 32, 128	findcard2s 9165	. 2 ⊢ (𝐴 ∈ Fin → (∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
130	129	imp 412	1 ⊢ ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⦋csb 3847   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ ciun 4951   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  Fincfn 8957  ℂcc 11179  ℝcr 11180  0cc0 11181   + caddc 11184   ≤ cle 11325  Σcsu 15833  vol*covol 25763

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9418  df-inf 9419  df-oi 9488  df-dju 9963  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-xadd 13223  df-ioo 13461  df-ico 13463  df-icc 13464  df-fz 13621  df-fzo 13769  df-fl 13912  df-seq 14125  df-exp 14185  df-hash 14455  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-clim 15635  df-sum 15834  df-xmet 21651  df-met 21652  df-ovol 25765

This theorem is used by:  volfiniun  25848  uniioombllem3a  25885  uniioombllem4  25887  i1fd  25982  i1fadd  25996  i1fmul  25997  volsupnfl  38551