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Theorem ovolfiniun 23789
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 3367 . . . 4 (𝑥 = ∅ → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4846 . . . . . 6 (𝑥 = ∅ → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
32fveq2d 6549 . . . . 5 (𝑥 = ∅ → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 14883 . . . . 5 (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 4981 . . . 4 (𝑥 = ∅ → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 346 . . 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 3367 . . . 4 (𝑥 = 𝑦 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4846 . . . . . 6 (𝑥 = 𝑦 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
98fveq2d 6549 . . . . 5 (𝑥 = 𝑦 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝑦 𝐵))
10 sumeq1 14883 . . . . 5 (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝑦 (vol*‘𝐵))
119, 10breq12d 4981 . . . 4 (𝑥 = 𝑦 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)))
127, 11imbi12d 346 . . 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 3367 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4846 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6549 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 14883 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 4981 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 346 . . 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 3367 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4846 . . . . . 6 (𝑥 = 𝐴 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
2120fveq2d 6549 . . . . 5 (𝑥 = 𝐴 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝐴 𝐵))
22 sumeq1 14883 . . . . 5 (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
2321, 22breq12d 4981 . . . 4 (𝑥 = 𝐴 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
2419, 23imbi12d 346 . . 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 11592 . . . . 5 0 ≤ 0
26 0iun 4891 . . . . . . 7 ∪ 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6548 . . . . . 6 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 23781 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2821 . . . . 5 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 14915 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 4998 . . . 4 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 4075 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 3960 . . . . . 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 767 . . . . . . . . . . . . 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 3839 . . . . . . . . . . . . . . . 16 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
39 nfcv 2951 . . . . . . . . . . . . . . . 16 Ⅎ𝑘ℝ
4038, 39nfss 3888 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ
41 nfcv 2951 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘vol*
4241, 38nffv 6555 . . . . . . . . . . . . . . . 16 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵)
4342nfel1 2965 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
4440, 43nfan 1885 . . . . . . . . . . . . . 14 Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
45 csbeq1a 3830 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
4645sseq1d 3925 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ))
4745fveq2d 6549 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
4847eleq1d 2869 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
4946, 48anbi12d 630 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5044, 49rspc 3555 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5137, 50mpan9 507 . . . . . . . . . . . 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 495 . . . . . . . . . . 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 3151 . . . . . . . . . 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 4874 . . . . . . . . . 10 (∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
5553, 54sylibr 235 . . . . . . . . 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 4844 . . . . . . . . . . . . 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 3906 . . . . . . . . . . 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 763 . . . . . . . . . . . 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 4079 . . . . . . . . . . . . 13 (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 496 . . . . . . . . . . . . 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 592 . . . . . . . . . . . 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 14928 . . . . . . . . . . 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 769 . . . . . . . . . . . 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 2951 . . . . . . . . . . . . . 14 Ⅎ𝑚𝐵
6665, 38, 45cbviun 4870 . . . . . . . . . . . . 13 ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
6766fveq2i 6548 . . . . . . . . . . . 12 (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
68 nfcv 2951 . . . . . . . . . . . . 13 Ⅎ𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 14891 . . . . . . . . . . . 12 Σ𝑘 ∈ 𝑦 (vol*‘𝐵) = Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵)
7064, 67, 693brtr3g 5001 . . . . . . . . . . 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 23771 . . . . . . . . . . 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 1364 . . . . . . . . . 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 4076 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4513 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3892 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3839 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
7776, 39nfss 3888 . . . . . . . . . . . . . 14 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ
7841, 76nffv 6555 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵)
7978nfel1 2965 . . . . . . . . . . . . . 14 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
8077, 79nfan 1885 . . . . . . . . . . . . 13 Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81 csbeq1a 3830 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
8281sseq1d 3925 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ))
8381fveq2d 6549 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
8483eleq1d 2869 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
8582, 84anbi12d 630 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
8680, 85rspc 3555 . . . . . . . . . . . 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 496 . . . . . . . . . 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 10523 . . . . . . . . 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 4921 . . . . . . . . . . . 12 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
91 vex 3443 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3820 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
9391, 92iunxsn 4918 . . . . . . . . . . . . 13 ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
9493uneq2i 4063 . . . . . . . . . . . 12 (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9590, 94eqtri 2821 . . . . . . . . . . 11 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9695fveq2i 6548 . . . . . . . . . 10 (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol*‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
97 ovolun 23787 . . . . . . . . . . 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 834 . . . . . . . . . 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 5003 . . . . . . . . 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 23771 . . . . . . . . 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 1364 . . . . . . . 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 8449 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 8638 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 687 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 722 . . . . . . . . 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 14928 . . . . . . . 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 11108 . . . . . . . . 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 765 . . . . . . . . . . . 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 4560 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
110108, 109sylibr 235 . . . . . . . . . . 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 2798 . . . . . . . . . . 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 10522 . . . . . . . . . . 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 14934 . . . . . . . . . 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 10522 . . . . . . . . . . . 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 6549 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
116115sumsn 14938 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
11791, 114, 116sylancr 587 . . . . . . . . . . 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 7039 . . . . . . . . . 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 2833 . . . . . . . . 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 4996 . . . . . . . 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 10650 . . . . . . 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 4870 . . . . . . . 8 ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
123122fveq2i 6548 . . . . . . 7 (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
12468, 42, 47cbvsumi 14891 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵)
125121, 123, 1243brtr4g 5002 . . . . . 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 421 . . . . 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 8612 . 2 (𝐴 ∈ Fin → (∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
130129imp 407 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 396   = wceq 1525   ∈ wcel 2083  ∀wral 3107  Vcvv 3440  ⦋csb 3817   ∪ cun 3863   ∩ cin 3864   ⊆ wss 3865  ∅c0 4217  {csn 4478  ∪ ciun 4831   class class class wbr 4968  ‘cfv 6232  (class class class)co 7023  Fincfn 8364  ℂcc 10388  ℝcr 10389  0cc0 10390   + caddc 10393   ≤ cle 10529  Σcsu 14880  vol*covol 23750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-inf2 8957  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467  ax-pre-sup 10468
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-int 4789  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-se 5410  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-isom 6241  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-of 7274  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-1o 7960  df-2o 7961  df-oadd 7964  df-er 8146  df-map 8265  df-en 8365  df-dom 8366  df-sdom 8367  df-fin 8368  df-sup 8759  df-inf 8760  df-oi 8827  df-dju 9183  df-card 9221  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-div 11152  df-nn 11493  df-2 11554  df-3 11555  df-n0 11752  df-z 11836  df-uz 12098  df-q 12202  df-rp 12244  df-xadd 12362  df-ioo 12596  df-ico 12598  df-icc 12599  df-fz 12747  df-fzo 12888  df-fl 13016  df-seq 13224  df-exp 13284  df-hash 13545  df-cj 14296  df-re 14297  df-im 14298  df-sqrt 14432  df-abs 14433  df-clim 14683  df-sum 14881  df-xmet 20224  df-met 20225  df-ovol 23752
This theorem is referenced by:  volfiniun  23835  uniioombllem3a  23872  uniioombllem4  23874  i1fd  23969  i1fadd  23983  i1fmul  23984  volsupnfl  34489
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