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Theorem ovolfiniun 23209
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 3131 . . . 4 (𝑥 = ∅ → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 4507 . . . . . 6 (𝑥 = ∅ → 𝑘𝑥 𝐵 = 𝑘 ∈ ∅ 𝐵)
32fveq2d 6162 . . . . 5 (𝑥 = ∅ → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 14369 . . . . 5 (𝑥 = ∅ → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 4636 . . . 4 (𝑥 = ∅ → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 334 . . 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 3131 . . . 4 (𝑥 = 𝑦 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 4507 . . . . . 6 (𝑥 = 𝑦 𝑘𝑥 𝐵 = 𝑘𝑦 𝐵)
98fveq2d 6162 . . . . 5 (𝑥 = 𝑦 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝑦 𝐵))
10 sumeq1 14369 . . . . 5 (𝑥 = 𝑦 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝑦 (vol*‘𝐵))
119, 10breq12d 4636 . . . 4 (𝑥 = 𝑦 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵)))
127, 11imbi12d 334 . . 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 3131 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 4507 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑘𝑥 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6162 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 14369 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 4636 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 334 . . 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 3131 . . . 4 (𝑥 = 𝐴 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 4507 . . . . . 6 (𝑥 = 𝐴 𝑘𝑥 𝐵 = 𝑘𝐴 𝐵)
2120fveq2d 6162 . . . . 5 (𝑥 = 𝐴 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝐴 𝐵))
22 sumeq1 14369 . . . . 5 (𝑥 = 𝐴 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝐴 (vol*‘𝐵))
2321, 22breq12d 4636 . . . 4 (𝑥 = 𝐴 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
2419, 23imbi12d 334 . . 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 11070 . . . . 5 0 ≤ 0
26 0iun 4550 . . . . . . 7 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6161 . . . . . 6 (vol*‘ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 23201 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2643 . . . . 5 (vol*‘ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 14401 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 4653 . . . 4 (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 3760 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 3651 . . . . . 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 793 . . . . . . . . . . . . 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 3535 . . . . . . . . . . . . . . . 16 𝑘𝑚 / 𝑘𝐵
39 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑘
4038, 39nfss 3581 . . . . . . . . . . . . . . 15 𝑘𝑚 / 𝑘𝐵 ⊆ ℝ
41 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑘vol*
4241, 38nffv 6165 . . . . . . . . . . . . . . . 16 𝑘(vol*‘𝑚 / 𝑘𝐵)
4342nfel1 2775 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑚 / 𝑘𝐵) ∈ ℝ
4440, 43nfan 1825 . . . . . . . . . . . . . 14 𝑘(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
45 csbeq1a 3528 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
4645sseq1d 3617 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ 𝑚 / 𝑘𝐵 ⊆ ℝ))
4745fveq2d 6162 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
4847eleq1d 2683 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
4946, 48anbi12d 746 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5044, 49rspc 3293 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5137, 50mpan9 486 . . . . . . . . . . . 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 475 . . . . . . . . . . 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 2962 . . . . . . . . . 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 4534 . . . . . . . . . 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 4505 . . . . . . . . . . . . 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 3599 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
59 simpll 789 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑦 ∈ Fin)
60 elun1 3764 . . . . . . . . . . . . 13 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 479 . . . . . . . . . . . . 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 491 . . . . . . . . . . . 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 14414 . . . . . . . . . . 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 795 . . . . . . . . . . . 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 2761 . . . . . . . . . . . . . 14 𝑚𝐵
6665, 38, 45cbviun 4530 . . . . . . . . . . . . 13 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
6766fveq2i 6161 . . . . . . . . . . . 12 (vol*‘ 𝑘𝑦 𝐵) = (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
68 nfcv 2761 . . . . . . . . . . . . 13 𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 14377 . . . . . . . . . . . 12 Σ𝑘𝑦 (vol*‘𝐵) = Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵)
7064, 67, 693brtr3g 4656 . . . . . . . . . . 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 23191 . . . . . . . . . . 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 1323 . . . . . . . . . 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 3761 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4187 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3585 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3535 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵
7776, 39nfss 3581 . . . . . . . . . . . . . 14 𝑘𝑧 / 𝑘𝐵 ⊆ ℝ
7841, 76nffv 6165 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑧 / 𝑘𝐵)
7978nfel1 2775 . . . . . . . . . . . . . 14 𝑘(vol*‘𝑧 / 𝑘𝐵) ∈ ℝ
8077, 79nfan 1825 . . . . . . . . . . . . 13 𝑘(𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3528 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281sseq1d 3617 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ 𝑧 / 𝑘𝐵 ⊆ ℝ))
8381fveq2d 6162 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
8483eleq1d 2683 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 746 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3293 . . . . . . . . . . . 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 479 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
8972, 88readdcld 10029 . . . . . . . . 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 4578 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
91 vex 3193 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3522 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
9391, 92iunxsn 4576 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
9493uneq2i 3748 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9590, 94eqtri 2643 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9695fveq2i 6161 . . . . . . . . . 10 (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol*‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
97 ovolun 23207 . . . . . . . . . . 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 1322 . . . . . . . . . 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 4658 . . . . . . . . 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 23191 . . . . . . . . 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 1323 . . . . . . . 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 7998 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 8187 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 706 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 761 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) ∈ Fin)
106105, 61fsumrecl 14414 . . . . . . . 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 10601 . . . . . . . . 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 791 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ¬ 𝑧𝑦)
109 disjsn 4223 . . . . . . . . . . . 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 2622 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
11261recnd 10028 . . . . . . . . . . 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 14420 . . . . . . . . . 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 10028 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ)
11592fveq2d 6162 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
116115sumsn 14424 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
11791, 114, 116sylancr 694 . . . . . . . . . . 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 6631 . . . . . . . . . 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 2655 . . . . . . . . 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 4651 . . . . . . . 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 10154 . . . . . . 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 4530 . . . . . . . 8 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
123122fveq2i 6161 . . . . . . 7 (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
12468, 42, 47cbvsumi 14377 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵)
125121, 123, 1243brtr4g 4657 . . . . . 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 8161 . 2 (𝐴 ∈ Fin → (∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
130129imp 445 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 1480  wcel 1987  wral 2908  Vcvv 3190  csb 3519  cun 3558  cin 3559  wss 3560  c0 3897  {csn 4155   ciun 4492   class class class wbr 4623  cfv 5857  (class class class)co 6615  Fincfn 7915  cc 9894  cr 9895  0cc0 9896   + caddc 9899  cle 10035  Σcsu 14366  vol*covol 23171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xadd 11907  df-ioo 12137  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-sum 14367  df-xmet 19679  df-met 19680  df-ovol 23173
This theorem is referenced by:  volfiniun  23255  uniioombllem3a  23292  uniioombllem4  23294  i1fd  23388  i1fadd  23402  i1fmul  23403  volsupnfl  33125
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