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Theorem ovolfiniun 25409
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 3298 . . . 4 (𝑥 = ∅ → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4975 . . . . . 6 (𝑥 = ∅ → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
32fveq2d 6865 . . . . 5 (𝑥 = ∅ → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 15662 . . . . 5 (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 5123 . . . 4 (𝑥 = ∅ → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 344 . . 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 3298 . . . 4 (𝑥 = 𝑦 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4975 . . . . . 6 (𝑥 = 𝑦 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
98fveq2d 6865 . . . . 5 (𝑥 = 𝑦 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝑦 𝐵))
10 sumeq1 15662 . . . . 5 (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝑦 (vol*‘𝐵))
119, 10breq12d 5123 . . . 4 (𝑥 = 𝑦 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)))
127, 11imbi12d 344 . . 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 3298 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4975 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6865 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 15662 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 5123 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 344 . . 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 3298 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4975 . . . . . 6 (𝑥 = 𝐴 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
2120fveq2d 6865 . . . . 5 (𝑥 = 𝐴 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝐴 𝐵))
22 sumeq1 15662 . . . . 5 (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
2321, 22breq12d 5123 . . . 4 (𝑥 = 𝐴 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
2419, 23imbi12d 344 . . 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 12294 . . . . 5 0 ≤ 0
26 0iun 5030 . . . . . . 7 ∪ 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6864 . . . . . 6 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 25401 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2753 . . . . 5 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 15694 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 5140 . . . 4 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 4144 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 4018 . . . . . 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 770 . . . . . . . . . . . . 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 3889 . . . . . . . . . . . . . . . 16 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
39 nfcv 2892 . . . . . . . . . . . . . . . 16 Ⅎ𝑘ℝ
4038, 39nfss 3942 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ
41 nfcv 2892 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘vol*
4241, 38nffv 6871 . . . . . . . . . . . . . . . 16 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵)
4342nfel1 2909 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
4440, 43nfan 1899 . . . . . . . . . . . . . 14 Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
45 csbeq1a 3879 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
4645sseq1d 3981 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ))
4745fveq2d 6865 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
4847eleq1d 2814 . . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5137, 50mpan9 506 . . . . . . . . . . . 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 494 . . . . . . . . . . 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 3126 . . . . . . . . . 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 5012 . . . . . . . . . 10 (∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
5553, 54sylibr 234 . . . . . . . . 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 4973 . . . . . . . . . . . . 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 3961 . . . . . . . . . . 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 766 . . . . . . . . . . . 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 4148 . . . . . . . . . . . . 13 (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 495 . . . . . . . . . . . . 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 593 . . . . . . . . . . . 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 15707 . . . . . . . . . . 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 772 . . . . . . . . . . . 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 2892 . . . . . . . . . . . . . 14 Ⅎ𝑚𝐵
6665, 38, 45cbviun 5003 . . . . . . . . . . . . 13 ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
6766fveq2i 6864 . . . . . . . . . . . 12 (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
68 nfcv 2892 . . . . . . . . . . . . 13 Ⅎ𝑚(vol*‘𝐵)
6947, 68, 42cbvsum 15668 . . . . . . . . . . . 12 Σ𝑘 ∈ 𝑦 (vol*‘𝐵) = Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵)
7064, 67, 693brtr3g 5143 . . . . . . . . . . 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 25391 . . . . . . . . . . 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 1373 . . . . . . . . . 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 4145 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4630 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3946 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3889 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
7776, 39nfss 3942 . . . . . . . . . . . . . 14 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ
7841, 76nffv 6871 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵)
7978nfel1 2909 . . . . . . . . . . . . . 14 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
8077, 79nfan 1899 . . . . . . . . . . . . 13 Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81 csbeq1a 3879 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
8281sseq1d 3981 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ))
8381fveq2d 6865 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
8483eleq1d 2814 . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . 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 495 . . . . . . . . . 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 11210 . . . . . . . . 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 5061 . . . . . . . . . . . 12 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
91 vex 3454 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3868 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
9391, 92iunxsn 5058 . . . . . . . . . . . . 13 ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
9493uneq2i 4131 . . . . . . . . . . . 12 (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9590, 94eqtri 2753 . . . . . . . . . . 11 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9695fveq2i 6864 . . . . . . . . . 10 (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol*‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
97 ovolun 25407 . . . . . . . . . . 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 837 . . . . . . . . . 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 5145 . . . . . . . . 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 25391 . . . . . . . . 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 1373 . . . . . . . 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 9017 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 9141 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 691 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 726 . . . . . . . . 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 15707 . . . . . . . 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 11799 . . . . . . . . 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 768 . . . . . . . . . . . 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 4678 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
110108, 109sylibr 234 . . . . . . . . . . 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 2731 . . . . . . . . . . 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 11209 . . . . . . . . . . 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 15714 . . . . . . . . . 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 11209 . . . . . . . . . . . 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 6865 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
116115sumsn 15719 . . . . . . . . . . . 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 7406 . . . . . . . . . 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 2765 . . . . . . . . 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 5138 . . . . . . . 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 11338 . . . . . . 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 5003 . . . . . . . 8 ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
123122fveq2i 6864 . . . . . . 7 (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
12447, 68, 42cbvsum 15668 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵)
125121, 123, 1243brtr4g 5144 . . . . . 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 420 . . . . 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 9135 . 2 (𝐴 ∈ Fin → (∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
130129imp 406 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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450  ⦋csb 3865   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  ∪ ciun 4958   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  ℂcc 11073  ℝcr 11074  0cc0 11075   + caddc 11078   ≤ cle 11216  Σcsu 15659  vol*covol 25370
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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-sup 9400  df-inf 9401  df-oi 9470  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xadd 13080  df-ioo 13317  df-ico 13319  df-icc 13320  df-fz 13476  df-fzo 13623  df-fl 13761  df-seq 13974  df-exp 14034  df-hash 14303  df-cj 15072  df-re 15073  df-im 15074  df-sqrt 15208  df-abs 15209  df-clim 15461  df-sum 15660  df-xmet 21264  df-met 21265  df-ovol 25372
This theorem is referenced by:  volfiniun  25455  uniioombllem3a  25492  uniioombllem4  25494  i1fd  25589  i1fadd  25603  i1fmul  25604  volsupnfl  37666
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