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Theorem ovolfiniun 23482
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 ∧ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (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 3327 . . . 4 (𝑥 = ∅ → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4726 . . . . . 6 (𝑥 = ∅ → 𝑘𝑥 𝐵 = 𝑘 ∈ ∅ 𝐵)
32fveq2d 6412 . . . . 5 (𝑥 = ∅ → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 14642 . . . . 5 (𝑥 = ∅ → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 4857 . . . 4 (𝑥 = ∅ → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 335 . . 3 (𝑥 = ∅ → ((∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵)) ↔ (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))))
7 raleq 3327 . . . 4 (𝑥 = 𝑦 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4726 . . . . . 6 (𝑥 = 𝑦 𝑘𝑥 𝐵 = 𝑘𝑦 𝐵)
98fveq2d 6412 . . . . 5 (𝑥 = 𝑦 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝑦 𝐵))
10 sumeq1 14642 . . . . 5 (𝑥 = 𝑦 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝑦 (vol*‘𝐵))
119, 10breq12d 4857 . . . 4 (𝑥 = 𝑦 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵)))
127, 11imbi12d 335 . . 3 (𝑥 = 𝑦 → ((∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵)) ↔ (∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))))
13 raleq 3327 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4726 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑘𝑥 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6412 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 14642 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 4857 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 335 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵)) ↔ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))))
19 raleq 3327 . . . 4 (𝑥 = 𝐴 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4726 . . . . . 6 (𝑥 = 𝐴 𝑘𝑥 𝐵 = 𝑘𝐴 𝐵)
2120fveq2d 6412 . . . . 5 (𝑥 = 𝐴 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝐴 𝐵))
22 sumeq1 14642 . . . . 5 (𝑥 = 𝐴 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝐴 (vol*‘𝐵))
2321, 22breq12d 4857 . . . 4 (𝑥 = 𝐴 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
2419, 23imbi12d 335 . . 3 (𝑥 = 𝐴 → ((∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵)) ↔ (∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵))))
25 0le0 11393 . . . . 5 0 ≤ 0
26 0iun 4769 . . . . . . 7 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6411 . . . . . 6 (vol*‘ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 23474 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2828 . . . . 5 (vol*‘ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 14675 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 4874 . . . 4 (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 3975 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 3863 . . . . . 6 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
3533, 34ax-mp 5 . . . . 5 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
3635imim1i 63 . . . 4 ((∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵)) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵)))
37 simprl 778 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ))
38 nfcsb1v 3744 . . . . . . . . . . . . . . . 16 𝑘𝑚 / 𝑘𝐵
39 nfcv 2948 . . . . . . . . . . . . . . . 16 𝑘
4038, 39nfss 3791 . . . . . . . . . . . . . . 15 𝑘𝑚 / 𝑘𝐵 ⊆ ℝ
41 nfcv 2948 . . . . . . . . . . . . . . . . 17 𝑘vol*
4241, 38nffv 6418 . . . . . . . . . . . . . . . 16 𝑘(vol*‘𝑚 / 𝑘𝐵)
4342nfel1 2963 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑚 / 𝑘𝐵) ∈ ℝ
4440, 43nfan 1990 . . . . . . . . . . . . . 14 𝑘(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
45 csbeq1a 3737 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
4645sseq1d 3829 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ 𝑚 / 𝑘𝐵 ⊆ ℝ))
4745fveq2d 6412 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
4847eleq1d 2870 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
4946, 48anbi12d 618 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5044, 49rspc 3496 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5137, 50mpan9 498 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
5251simpld 484 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
5352ralrimiva 3154 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ)
54 iunss 4753 . . . . . . . . . 10 ( 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ)
5553, 54sylibr 225 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ)
56 iunss1 4724 . . . . . . . . . . . . 13 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
5733, 56ax-mp 5 . . . . . . . . . . . 12 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
5857, 55syl5ss 3809 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
59 simpll 774 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑦 ∈ Fin)
60 elun1 3979 . . . . . . . . . . . . 13 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 485 . . . . . . . . . . . . 13 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
6260, 61sylan2 582 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) ∧ 𝑚𝑦) → (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
6359, 62fsumrecl 14688 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
64 simprr 780 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))
65 nfcv 2948 . . . . . . . . . . . . . 14 𝑚𝐵
6665, 38, 45cbviun 4749 . . . . . . . . . . . . 13 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
6766fveq2i 6411 . . . . . . . . . . . 12 (vol*‘ 𝑘𝑦 𝐵) = (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
68 nfcv 2948 . . . . . . . . . . . . 13 𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 14650 . . . . . . . . . . . 12 Σ𝑘𝑦 (vol*‘𝐵) = Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵)
7064, 67, 693brtr3g 4877 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
71 ovollecl 23464 . . . . . . . . . . 11 (( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ∧ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ ∧ (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
7258, 63, 70, 71syl3anc 1483 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
73 ssun2 3976 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4403 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3795 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3744 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵
7776, 39nfss 3791 . . . . . . . . . . . . . 14 𝑘𝑧 / 𝑘𝐵 ⊆ ℝ
7841, 76nffv 6418 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑧 / 𝑘𝐵)
7978nfel1 2963 . . . . . . . . . . . . . 14 𝑘(vol*‘𝑧 / 𝑘𝐵) ∈ ℝ
8077, 79nfan 1990 . . . . . . . . . . . . 13 𝑘(𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3737 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281sseq1d 3829 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ 𝑧 / 𝑘𝐵 ⊆ ℝ))
8381fveq2d 6412 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
8483eleq1d 2870 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 618 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3496 . . . . . . . . . . . 12 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8775, 37, 86mpsyl 68 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ))
8887simprd 485 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
8972, 88readdcld 10354 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)) ∈ ℝ)
90 iunxun 4797 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
91 vex 3394 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3731 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
9391, 92iunxsn 4795 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
9493uneq2i 3963 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9590, 94eqtri 2828 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9695fveq2i 6411 . . . . . . . . . 10 (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol*‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
97 ovolun 23480 . . . . . . . . . . 11 ((( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ) ∧ (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)) → (vol*‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)) ≤ ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)))
9858, 72, 87, 97syl21anc 857 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)) ≤ ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)))
9996, 98syl5eqbr 4879 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) ≤ ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)))
100 ovollecl 23464 . . . . . . . . 9 (( 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ ∧ ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)) ∈ ℝ ∧ (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) ≤ ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵))) → (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) ∈ ℝ)
10155, 89, 99, 100syl3anc 1483 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) ∈ ℝ)
102 snfi 8277 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 8466 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 674 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 708 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) ∈ Fin)
106105, 61fsumrecl 14688 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
10772, 63, 88, 70leadd1dd 10926 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)) ≤ (Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)))
108 simplr 776 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ¬ 𝑧𝑦)
109 disjsn 4438 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
110108, 109sylibr 225 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∩ {𝑧}) = ∅)
111 eqidd 2807 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
11261recnd 10353 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol*‘𝑚 / 𝑘𝐵) ∈ ℂ)
113110, 111, 105, 112fsumsplit 14694 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵) = (Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) + Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵)))
11488recnd 10353 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ)
11592fveq2d 6412 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
116115sumsn 14698 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
11791, 114, 116sylancr 577 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
118117oveq2d 6890 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) + Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵)) = (Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)))
119113, 118eqtrd 2840 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵) = (Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)))
120107, 119breqtrrd 4872 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ((vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol*‘𝑧 / 𝑘𝐵)) ≤ Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵))
121101, 89, 106, 99, 120letrd 10479 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) ≤ Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵))
12265, 38, 45cbviun 4749 . . . . . . . 8 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
123122fveq2i 6411 . . . . . . 7 (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
12468, 42, 47cbvsumi 14650 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵)
125121, 123, 1243brtr4g 4878 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
126125exp32 409 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵) → (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))))
127126a2d 29 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (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 ∧ ¬ 𝑧𝑦) → ((∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (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 8440 . 2 (𝐴 ∈ Fin → (∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
130129imp 395 1 ((𝐴 ∈ Fin ∧ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1637  wcel 2156  wral 3096  Vcvv 3391  csb 3728  cun 3767  cin 3768  wss 3769  c0 4116  {csn 4370   ciun 4712   class class class wbr 4844  cfv 6101  (class class class)co 6874  Fincfn 8192  cc 10219  cr 10220  0cc0 10221   + caddc 10224  cle 10360  Σcsu 14639  vol*covol 23443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-inf2 8785  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298  ax-pre-sup 10299
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-of 7127  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-2o 7797  df-oadd 7800  df-er 7979  df-map 8094  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-sup 8587  df-inf 8588  df-oi 8654  df-card 9048  df-cda 9275  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-div 10970  df-nn 11306  df-2 11364  df-3 11365  df-n0 11560  df-z 11644  df-uz 11905  df-q 12008  df-rp 12047  df-xadd 12163  df-ioo 12397  df-ico 12399  df-icc 12400  df-fz 12550  df-fzo 12690  df-fl 12817  df-seq 13025  df-exp 13084  df-hash 13338  df-cj 14062  df-re 14063  df-im 14064  df-sqrt 14198  df-abs 14199  df-clim 14442  df-sum 14640  df-xmet 19947  df-met 19948  df-ovol 23445
This theorem is referenced by:  volfiniun  23528  uniioombllem3a  23565  uniioombllem4  23567  i1fd  23662  i1fadd  23676  i1fmul  23677  volsupnfl  33767
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