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Theorem ovolfiniun 23315
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 3168 . . . 4 (𝑥 = ∅ → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4566 . . . . . 6 (𝑥 = ∅ → 𝑘𝑥 𝐵 = 𝑘 ∈ ∅ 𝐵)
32fveq2d 6233 . . . . 5 (𝑥 = ∅ → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 14463 . . . . 5 (𝑥 = ∅ → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 4698 . . . 4 (𝑥 = ∅ → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 333 . . 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 3168 . . . 4 (𝑥 = 𝑦 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4566 . . . . . 6 (𝑥 = 𝑦 𝑘𝑥 𝐵 = 𝑘𝑦 𝐵)
98fveq2d 6233 . . . . 5 (𝑥 = 𝑦 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝑦 𝐵))
10 sumeq1 14463 . . . . 5 (𝑥 = 𝑦 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝑦 (vol*‘𝐵))
119, 10breq12d 4698 . . . 4 (𝑥 = 𝑦 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵)))
127, 11imbi12d 333 . . 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 3168 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4566 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑘𝑥 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6233 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 14463 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 4698 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 333 . . 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 3168 . . . 4 (𝑥 = 𝐴 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4566 . . . . . 6 (𝑥 = 𝐴 𝑘𝑥 𝐵 = 𝑘𝐴 𝐵)
2120fveq2d 6233 . . . . 5 (𝑥 = 𝐴 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝐴 𝐵))
22 sumeq1 14463 . . . . 5 (𝑥 = 𝐴 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝐴 (vol*‘𝐵))
2321, 22breq12d 4698 . . . 4 (𝑥 = 𝐴 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
2419, 23imbi12d 333 . . 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 11148 . . . . 5 0 ≤ 0
26 0iun 4609 . . . . . . 7 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6232 . . . . . 6 (vol*‘ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 23307 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2673 . . . . 5 (vol*‘ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 14496 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 4715 . . . 4 (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 3809 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 3699 . . . . . 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 809 . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . 16 𝑘𝑚 / 𝑘𝐵
39 nfcv 2793 . . . . . . . . . . . . . . . 16 𝑘
4038, 39nfss 3629 . . . . . . . . . . . . . . 15 𝑘𝑚 / 𝑘𝐵 ⊆ ℝ
41 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑘vol*
4241, 38nffv 6236 . . . . . . . . . . . . . . . 16 𝑘(vol*‘𝑚 / 𝑘𝐵)
4342nfel1 2808 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑚 / 𝑘𝐵) ∈ ℝ
4440, 43nfan 1868 . . . . . . . . . . . . . 14 𝑘(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
45 csbeq1a 3575 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
4645sseq1d 3665 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ 𝑚 / 𝑘𝐵 ⊆ ℝ))
4745fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
4847eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
4946, 48anbi12d 747 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5044, 49rspc 3334 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5137, 50mpan9 485 . . . . . . . . . . . 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 474 . . . . . . . . . . 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 2995 . . . . . . . . . 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 4593 . . . . . . . . . 10 ( 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ)
5553, 54sylibr 224 . . . . . . . . 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 4564 . . . . . . . . . . . . 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 3647 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
59 simpll 805 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑦 ∈ Fin)
60 elun1 3813 . . . . . . . . . . . . 13 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 478 . . . . . . . . . . . . 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 490 . . . . . . . . . . . 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 14509 . . . . . . . . . . 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 811 . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . 14 𝑚𝐵
6665, 38, 45cbviun 4589 . . . . . . . . . . . . 13 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
6766fveq2i 6232 . . . . . . . . . . . 12 (vol*‘ 𝑘𝑦 𝐵) = (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
68 nfcv 2793 . . . . . . . . . . . . 13 𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 14471 . . . . . . . . . . . 12 Σ𝑘𝑦 (vol*‘𝐵) = Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵)
7064, 67, 693brtr3g 4718 . . . . . . . . . . 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 23297 . . . . . . . . . . 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 1366 . . . . . . . . . 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 3810 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4242 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3633 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3582 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵
7776, 39nfss 3629 . . . . . . . . . . . . . 14 𝑘𝑧 / 𝑘𝐵 ⊆ ℝ
7841, 76nffv 6236 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑧 / 𝑘𝐵)
7978nfel1 2808 . . . . . . . . . . . . . 14 𝑘(vol*‘𝑧 / 𝑘𝐵) ∈ ℝ
8077, 79nfan 1868 . . . . . . . . . . . . 13 𝑘(𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3575 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281sseq1d 3665 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ 𝑧 / 𝑘𝐵 ⊆ ℝ))
8381fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
8483eleq1d 2715 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 747 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3334 . . . . . . . . . . . 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 478 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
8972, 88readdcld 10107 . . . . . . . . 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 4637 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
91 vex 3234 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3569 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
9391, 92iunxsn 4635 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
9493uneq2i 3797 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9590, 94eqtri 2673 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9695fveq2i 6232 . . . . . . . . . 10 (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol*‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
97 ovolun 23313 . . . . . . . . . . 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 1365 . . . . . . . . . 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 4720 . . . . . . . . 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 23297 . . . . . . . . 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 1366 . . . . . . . 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 8079 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 8268 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 707 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 762 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) ∈ Fin)
106105, 61fsumrecl 14509 . . . . . . . 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 10679 . . . . . . . . 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 807 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ¬ 𝑧𝑦)
109 disjsn 4278 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
110108, 109sylibr 224 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∩ {𝑧}) = ∅)
111 eqidd 2652 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
11261recnd 10106 . . . . . . . . . . 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 14515 . . . . . . . . . 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 10106 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ)
11592fveq2d 6233 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
116115sumsn 14519 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
11791, 114, 116sylancr 696 . . . . . . . . . . 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 6706 . . . . . . . . . 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 2685 . . . . . . . . 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 4713 . . . . . . . 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 10232 . . . . . . 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 4589 . . . . . . . 8 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
123122fveq2i 6232 . . . . . . 7 (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
12468, 42, 47cbvsumi 14471 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵)
125121, 123, 1243brtr4g 4719 . . . . . 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 630 . . . . 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 8242 . 2 (𝐴 ∈ Fin → (∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
130129imp 444 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 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  csb 3566  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   ciun 4552   class class class wbr 4685  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  cr 9973  0cc0 9974   + caddc 9977  cle 10113  Σcsu 14460  vol*covol 23277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279
This theorem is referenced by:  volfiniun  23361  uniioombllem3a  23398  uniioombllem4  23400  i1fd  23493  i1fadd  23507  i1fmul  23508  volsupnfl  33584
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