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Theorem ovolfiniun 24710
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.
StepHypRef Expression
1 raleq 3354 . . . 4 (𝑥 = ∅ → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4947 . . . . . 6 (𝑥 = ∅ → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
32fveq2d 6808 . . . . 5 (𝑥 = ∅ → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 15445 . . . . 5 (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 5094 . . . 4 (𝑥 = ∅ → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 345 . . 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 3354 . . . 4 (𝑥 = 𝑦 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4947 . . . . . 6 (𝑥 = 𝑦 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
98fveq2d 6808 . . . . 5 (𝑥 = 𝑦 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝑦 𝐵))
10 sumeq1 15445 . . . . 5 (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝑦 (vol*‘𝐵))
119, 10breq12d 5094 . . . 4 (𝑥 = 𝑦 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)))
127, 11imbi12d 345 . . 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 3354 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4947 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6808 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 15445 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 5094 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 345 . . 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 3354 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4947 . . . . . 6 (𝑥 = 𝐴 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
2120fveq2d 6808 . . . . 5 (𝑥 = 𝐴 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝐴 𝐵))
22 sumeq1 15445 . . . . 5 (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
2321, 22breq12d 5094 . . . 4 (𝑥 = 𝐴 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
2419, 23imbi12d 345 . . 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 12120 . . . . 5 0 ≤ 0
26 0iun 4999 . . . . . . 7 ∪ 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6807 . . . . . 6 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 24702 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2764 . . . . 5 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 15478 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 5111 . . . 4 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 4112 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 3992 . . . . . 6 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
3533, 34ax-mp 5 . . . . 5 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
3635imim1i 63 . . . 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 769 . . . . . . . . . . . . 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 3862 . . . . . . . . . . . . . . . 16 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
39 nfcv 2905 . . . . . . . . . . . . . . . 16 Ⅎ𝑘ℝ
4038, 39nfss 3918 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ
41 nfcv 2905 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘vol*
4241, 38nffv 6814 . . . . . . . . . . . . . . . 16 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵)
4342nfel1 2921 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
4440, 43nfan 1900 . . . . . . . . . . . . . 14 Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
45 csbeq1a 3851 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
4645sseq1d 3957 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ))
4745fveq2d 6808 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
4847eleq1d 2821 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
4946, 48anbi12d 632 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5044, 49rspc 3554 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5137, 50mpan9 508 . . . . . . . . . . . 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))
5251simpld 496 . . . . . . . . . . 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)
5352ralrimiva 3140 . . . . . . . . . 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 4982 . . . . . . . . . 10 (∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
5553, 54sylibr 233 . . . . . . . . 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 4945 . . . . . . . . . . . . 13 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
5733, 56ax-mp 5 . . . . . . . . . . . 12 ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ⊆ ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
5857, 55sstrid 3937 . . . . . . . . . . 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 765 . . . . . . . . . . . 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 4116 . . . . . . . . . . . . 13 (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 497 . . . . . . . . . . . . 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)
6260, 61sylan2 594 . . . . . . . . . . . 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)
6359, 62fsumrecl 15491 . . . . . . . . . . 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 771 . . . . . . . . . . . 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 2905 . . . . . . . . . . . . . 14 Ⅎ𝑚𝐵
6665, 38, 45cbviun 4973 . . . . . . . . . . . . 13 ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
6766fveq2i 6807 . . . . . . . . . . . 12 (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
68 nfcv 2905 . . . . . . . . . . . . 13 Ⅎ𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 15454 . . . . . . . . . . . 12 Σ𝑘 ∈ 𝑦 (vol*‘𝐵) = Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵)
7064, 67, 693brtr3g 5114 . . . . . . . . . . 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 24692 . . . . . . . . . . 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)
7258, 63, 70, 71syl3anc 1371 . . . . . . . . . 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 4113 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4602 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3923 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3862 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
7776, 39nfss 3918 . . . . . . . . . . . . . 14 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ
7841, 76nffv 6814 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵)
7978nfel1 2921 . . . . . . . . . . . . . 14 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
8077, 79nfan 1900 . . . . . . . . . . . . 13 Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81 csbeq1a 3851 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
8281sseq1d 3957 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ))
8381fveq2d 6808 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
8483eleq1d 2821 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
8582, 84anbi12d 632 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
8680, 85rspc 3554 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
8775, 37, 86mpsyl 68 . . . . . . . . . . 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))
8887simprd 497 . . . . . . . . . 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)
8972, 88readdcld 11050 . . . . . . . . 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 5030 . . . . . . . . . . . 12 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
91 vex 3441 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3840 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
9391, 92iunxsn 5027 . . . . . . . . . . . . 13 ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
9493uneq2i 4100 . . . . . . . . . . . 12 (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9590, 94eqtri 2764 . . . . . . . . . . 11 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9695fveq2i 6807 . . . . . . . . . 10 (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol*‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
97 ovolun 24708 . . . . . . . . . . 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)))
9858, 72, 87, 97syl21anc 836 . . . . . . . . . 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)))
9996, 98eqbrtrid 5116 . . . . . . . . 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 24692 . . . . . . . . 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)
10155, 89, 99, 100syl3anc 1371 . . . . . . . 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 8869 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 8993 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 689 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 724 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) ∈ Fin)
106105, 61fsumrecl 15491 . . . . . . . 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)
10772, 63, 88, 70leadd1dd 11635 . . . . . . . . 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 767 . . . . . . . . . . . 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 4651 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
110108, 109sylibr 233 . . . . . . . . . . 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 2737 . . . . . . . . . . 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}))
11261recnd 11049 . . . . . . . . . . 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)
113110, 111, 105, 112fsumsplit 15498 . . . . . . . . . 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)))
11488recnd 11049 . . . . . . . . . . . 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)
11592fveq2d 6808 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
116115sumsn 15503 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
11791, 114, 116sylancr 588 . . . . . . . . . . 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))
118117oveq2d 7323 . . . . . . . . . 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)))
119113, 118eqtrd 2776 . . . . . . . . 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)))
120107, 119breqtrrd 5109 . . . . . . . 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))
121101, 89, 106, 99, 120letrd 11178 . . . . . . 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))
12265, 38, 45cbviun 4973 . . . . . . . 8 ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
123122fveq2i 6807 . . . . . . 7 (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
12468, 42, 47cbvsumi 15454 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵)
125121, 123, 1243brtr4g 5115 . . . . . 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))
126125exp32 422 . . . . 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))))
127126a2d 29 . . . 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))))
12836, 127syl5 34 . . 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))))
1296, 12, 18, 24, 32, 128findcard2s 8986 . 2 (𝐴 ∈ Fin → (∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
130129imp 408 1 ((𝐴 ∈ Fin ∧ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  Vcvv 3437  ⦋csb 3837   ∪ cun 3890   ∩ cin 3891   ⊆ wss 3892  ∅c0 4262  {csn 4565  ∪ ciun 4931   class class class wbr 5081  ‘cfv 6458  (class class class)co 7307  Fincfn 8764  ℂcc 10915  ℝcr 10916  0cc0 10917   + caddc 10920   ≤ cle 11056  Σcsu 15442  vol*covol 24671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-inf2 9443  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994  ax-pre-sup 10995
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-se 5556  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-isom 6467  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-of 7565  df-om 7745  df-1st 7863  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-2o 8329  df-er 8529  df-map 8648  df-en 8765  df-dom 8766  df-sdom 8767  df-fin 8768  df-sup 9245  df-inf 9246  df-oi 9313  df-dju 9703  df-card 9741  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-div 11679  df-nn 12020  df-2 12082  df-3 12083  df-n0 12280  df-z 12366  df-uz 12629  df-q 12735  df-rp 12777  df-xadd 12895  df-ioo 13129  df-ico 13131  df-icc 13132  df-fz 13286  df-fzo 13429  df-fl 13558  df-seq 13768  df-exp 13829  df-hash 14091  df-cj 14855  df-re 14856  df-im 14857  df-sqrt 14991  df-abs 14992  df-clim 15242  df-sum 15443  df-xmet 20635  df-met 20636  df-ovol 24673
This theorem is referenced by:  volfiniun  24756  uniioombllem3a  24793  uniioombllem4  24795  i1fd  24890  i1fadd  24904  i1fmul  24905  volsupnfl  35866
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