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Theorem ovolfiniun 23269
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 3138 . . . 4 (𝑥 = ∅ → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4534 . . . . . 6 (𝑥 = ∅ → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵)
32fveq2d 6195 . . . . 5 (𝑥 = ∅ → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 14419 . . . . 5 (𝑥 = ∅ → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 4666 . . . 4 (𝑥 = ∅ → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 334 . . 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 3138 . . . 4 (𝑥 = 𝑦 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4534 . . . . . 6 (𝑥 = 𝑦 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝑦 𝐵)
98fveq2d 6195 . . . . 5 (𝑥 = 𝑦 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝑦 𝐵))
10 sumeq1 14419 . . . . 5 (𝑥 = 𝑦 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝑦 (vol*‘𝐵))
119, 10breq12d 4666 . . . 4 (𝑥 = 𝑦 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) ≤ Σ𝑘 ∈ 𝑦 (vol*‘𝐵)))
127, 11imbi12d 334 . . 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 3138 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4534 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6195 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 14419 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 4666 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 334 . . 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 3138 . . . 4 (𝑥 = 𝐴 → (∀𝑘 ∈ 𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4534 . . . . . 6 (𝑥 = 𝐴 → ∪ 𝑘 ∈ 𝑥 𝐵 = ∪ 𝑘 ∈ 𝐴 𝐵)
2120fveq2d 6195 . . . . 5 (𝑥 = 𝐴 → (vol*‘∪ 𝑘 ∈ 𝑥 𝐵) = (vol*‘∪ 𝑘 ∈ 𝐴 𝐵))
22 sumeq1 14419 . . . . 5 (𝑥 = 𝐴 → Σ𝑘 ∈ 𝑥 (vol*‘𝐵) = Σ𝑘 ∈ 𝐴 (vol*‘𝐵))
2321, 22breq12d 4666 . . . 4 (𝑥 = 𝐴 → ((vol*‘∪ 𝑘 ∈ 𝑥 𝐵) ≤ Σ𝑘 ∈ 𝑥 (vol*‘𝐵) ↔ (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
2419, 23imbi12d 334 . . 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 11110 . . . . 5 0 ≤ 0
26 0iun 4577 . . . . . . 7 ∪ 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6194 . . . . . 6 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 23261 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2644 . . . . 5 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 14452 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 4683 . . . 4 (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 3776 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 3666 . . . . . 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 794 . . . . . . . . . . . . 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 3549 . . . . . . . . . . . . . . . 16 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵
39 nfcv 2764 . . . . . . . . . . . . . . . 16 Ⅎ𝑘ℝ
4038, 39nfss 3596 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ
41 nfcv 2764 . . . . . . . . . . . . . . . . 17 Ⅎ𝑘vol*
4241, 38nffv 6198 . . . . . . . . . . . . . . . 16 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵)
4342nfel1 2779 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ
4440, 43nfan 1828 . . . . . . . . . . . . . 14 Ⅎ𝑘(⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)
45 csbeq1a 3542 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑘⦌𝐵)
4645sseq1d 3632 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ ⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ))
4745fveq2d 6195 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘⦋𝑚 / 𝑘⦌𝐵))
4847eleq1d 2686 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ))
4946, 48anbi12d 747 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5044, 49rspc 3303 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑚 / 𝑘⦌𝐵) ∈ ℝ)))
5137, 50mpan9 486 . . . . . . . . . . . 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 475 . . . . . . . . . . 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 2966 . . . . . . . . . 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 4561 . . . . . . . . . 10 (∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 ⊆ ℝ)
5553, 54sylibr 224 . . . . . . . . 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 4532 . . . . . . . . . . . . 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, 55syl5ss 3614 . . . . . . . . . . 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 790 . . . . . . . . . . . 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 3780 . . . . . . . . . . . . 13 (𝑚 ∈ 𝑦 → 𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 479 . . . . . . . . . . . . 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 491 . . . . . . . . . . . 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 14465 . . . . . . . . . . 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 796 . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . 14 Ⅎ𝑚𝐵
6665, 38, 45cbviun 4557 . . . . . . . . . . . . 13 ∪ 𝑘 ∈ 𝑦 𝐵 = ∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵
6766fveq2i 6194 . . . . . . . . . . . 12 (vol*‘∪ 𝑘 ∈ 𝑦 𝐵) = (vol*‘∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵)
68 nfcv 2764 . . . . . . . . . . . . 13 Ⅎ𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 14427 . . . . . . . . . . . 12 Σ𝑘 ∈ 𝑦 (vol*‘𝐵) = Σ𝑚 ∈ 𝑦 (vol*‘⦋𝑚 / 𝑘⦌𝐵)
7064, 67, 693brtr3g 4686 . . . . . . . . . . 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 23251 . . . . . . . . . . 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 1326 . . . . . . . . . 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 3777 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4209 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3600 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3549 . . . . . . . . . . . . . . 15 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵
7776, 39nfss 3596 . . . . . . . . . . . . . 14 Ⅎ𝑘⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ
7841, 76nffv 6198 . . . . . . . . . . . . . . 15 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵)
7978nfel1 2779 . . . . . . . . . . . . . 14 Ⅎ𝑘(vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ
8077, 79nfan 1828 . . . . . . . . . . . . 13 Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)
81 csbeq1a 3542 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
8281sseq1d 3632 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ ⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ))
8381fveq2d 6195 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
8483eleq1d 2686 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ))
8582, 84anbi12d 747 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (⦋𝑧 / 𝑘⦌𝐵 ⊆ ℝ ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℝ)))
8680, 85rspc 3303 . . . . . . . . . . . 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 479 . . . . . . . . . 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 10069 . . . . . . . . 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 4605 . . . . . . . . . . . 12 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵)
91 vex 3203 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3536 . . . . . . . . . . . . . 14 (𝑚 = 𝑧 → ⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵)
9391, 92iunxsn 4603 . . . . . . . . . . . . 13 ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵 = ⦋𝑧 / 𝑘⦌𝐵
9493uneq2i 3764 . . . . . . . . . . . 12 (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ∪ 𝑚 ∈ {𝑧}⦋𝑚 / 𝑘⦌𝐵) = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9590, 94eqtri 2644 . . . . . . . . . . 11 ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵 = (∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵)
9695fveq2i 6194 . . . . . . . . . 10 (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵) = (vol*‘(∪ 𝑚 ∈ 𝑦 ⦋𝑚 / 𝑘⦌𝐵 ∪ ⦋𝑧 / 𝑘⦌𝐵))
97 ovolun 23267 . . . . . . . . . . 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 1325 . . . . . . . . . 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, 98syl5eqbr 4688 . . . . . . . . 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 23251 . . . . . . . . 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 1326 . . . . . . . 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 8038 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 8227 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 707 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 762 . . . . . . . . 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 14465 . . . . . . . 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 10641 . . . . . . . . 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 792 . . . . . . . . . . . 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 4246 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
110108, 109sylibr 224 . . . . . . . . . . 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 2623 . . . . . . . . . . 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 10068 . . . . . . . . . . 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 14471 . . . . . . . . . 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 10068 . . . . . . . . . . . 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 6195 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
116115sumsn 14475 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘⦋𝑧 / 𝑘⦌𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘⦋𝑚 / 𝑘⦌𝐵) = (vol*‘⦋𝑧 / 𝑘⦌𝐵))
11791, 114, 116sylancr 695 . . . . . . . . . . 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 6666 . . . . . . . . . 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 2656 . . . . . . . . 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 4681 . . . . . . . 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 10194 . . . . . . 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 4557 . . . . . . . 8 ∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = ∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵
123122fveq2i 6194 . . . . . . 7 (vol*‘∪ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘∪ 𝑚 ∈ (𝑦 ∪ {𝑧})⦋𝑚 / 𝑘⦌𝐵)
12468, 42, 47cbvsumi 14427 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘⦋𝑚 / 𝑘⦌𝐵)
125121, 123, 1243brtr4g 4687 . . . . . 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 631 . . . . 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 8201 . 2 (𝐴 ∈ Fin → (∀𝑘 ∈ 𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘∪ 𝑘 ∈ 𝐴 𝐵) ≤ Σ𝑘 ∈ 𝐴 (vol*‘𝐵)))
130129imp 445 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 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200  ⦋csb 3533   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ ciun 4520   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936   + caddc 9939   ≤ cle 10075  Σcsu 14416  vol*covol 23231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xadd 11947  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-xmet 19739  df-met 19740  df-ovol 23233
This theorem is referenced by:  volfiniun  23315  uniioombllem3a  23352  uniioombllem4  23354  i1fd  23448  i1fadd  23462  i1fmul  23463  volsupnfl  33454
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