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Theorem ovolfiniun 25521
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 3312 . . . 4 (𝑥 = ∅ → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
2 iuneq1 5017 . . . . . 6 (𝑥 = ∅ → 𝑘𝑥 𝐵 = 𝑘 ∈ ∅ 𝐵)
32fveq2d 6905 . . . . 5 (𝑥 = ∅ → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ ∅ 𝐵))
4 sumeq1 15693 . . . . 5 (𝑥 = ∅ → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ ∅ (vol*‘𝐵))
53, 4breq12d 5166 . . . 4 (𝑥 = ∅ → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)))
61, 5imbi12d 343 . . 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 3312 . . . 4 (𝑥 = 𝑦 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
8 iuneq1 5017 . . . . . 6 (𝑥 = 𝑦 𝑘𝑥 𝐵 = 𝑘𝑦 𝐵)
98fveq2d 6905 . . . . 5 (𝑥 = 𝑦 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝑦 𝐵))
10 sumeq1 15693 . . . . 5 (𝑥 = 𝑦 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝑦 (vol*‘𝐵))
119, 10breq12d 5166 . . . 4 (𝑥 = 𝑦 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵)))
127, 11imbi12d 343 . . 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 3312 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
14 iuneq1 5017 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → 𝑘𝑥 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
1514fveq2d 6905 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
16 sumeq1 15693 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵))
1715, 16breq12d 5166 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) ≤ Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵)))
1813, 17imbi12d 343 . . 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 3312 . . . 4 (𝑥 = 𝐴 → (∀𝑘𝑥 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)))
20 iuneq1 5017 . . . . . 6 (𝑥 = 𝐴 𝑘𝑥 𝐵 = 𝑘𝐴 𝐵)
2120fveq2d 6905 . . . . 5 (𝑥 = 𝐴 → (vol*‘ 𝑘𝑥 𝐵) = (vol*‘ 𝑘𝐴 𝐵))
22 sumeq1 15693 . . . . 5 (𝑥 = 𝐴 → Σ𝑘𝑥 (vol*‘𝐵) = Σ𝑘𝐴 (vol*‘𝐵))
2321, 22breq12d 5166 . . . 4 (𝑥 = 𝐴 → ((vol*‘ 𝑘𝑥 𝐵) ≤ Σ𝑘𝑥 (vol*‘𝐵) ↔ (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
2419, 23imbi12d 343 . . 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 12365 . . . . 5 0 ≤ 0
26 0iun 5071 . . . . . . 7 𝑘 ∈ ∅ 𝐵 = ∅
2726fveq2i 6904 . . . . . 6 (vol*‘ 𝑘 ∈ ∅ 𝐵) = (vol*‘∅)
28 ovol0 25513 . . . . . 6 (vol*‘∅) = 0
2927, 28eqtri 2754 . . . . 5 (vol*‘ 𝑘 ∈ ∅ 𝐵) = 0
30 sum0 15725 . . . . 5 Σ𝑘 ∈ ∅ (vol*‘𝐵) = 0
3125, 29, 303brtr4i 5183 . . . 4 (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵)
3231a1i 11 . . 3 (∀𝑘 ∈ ∅ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘 ∈ ∅ 𝐵) ≤ Σ𝑘 ∈ ∅ (vol*‘𝐵))
33 ssun1 4173 . . . . . 6 𝑦 ⊆ (𝑦 ∪ {𝑧})
34 ssralv 4048 . . . . . 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 769 . . . . . . . . . . . . 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 3917 . . . . . . . . . . . . . . . 16 𝑘𝑚 / 𝑘𝐵
39 nfcv 2892 . . . . . . . . . . . . . . . 16 𝑘
4038, 39nfss 3972 . . . . . . . . . . . . . . 15 𝑘𝑚 / 𝑘𝐵 ⊆ ℝ
41 nfcv 2892 . . . . . . . . . . . . . . . . 17 𝑘vol*
4241, 38nffv 6911 . . . . . . . . . . . . . . . 16 𝑘(vol*‘𝑚 / 𝑘𝐵)
4342nfel1 2909 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑚 / 𝑘𝐵) ∈ ℝ
4440, 43nfan 1895 . . . . . . . . . . . . . 14 𝑘(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
45 csbeq1a 3906 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
4645sseq1d 4011 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → (𝐵 ⊆ ℝ ↔ 𝑚 / 𝑘𝐵 ⊆ ℝ))
4745fveq2d 6905 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑚 → (vol*‘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
4847eleq1d 2811 . . . . . . . . . . . . . . 15 (𝑘 = 𝑚 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
4946, 48anbi12d 630 . . . . . . . . . . . . . 14 (𝑘 = 𝑚 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5044, 49rspc 3596 . . . . . . . . . . . . 13 (𝑚 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
5137, 50mpan9 505 . . . . . . . . . . . 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 493 . . . . . . . . . . 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 3136 . . . . . . . . . 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 5053 . . . . . . . . . 10 ( 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 ⊆ ℝ)
5553, 54sylibr 233 . . . . . . . . 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 5015 . . . . . . . . . . . . 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, 55sstrid 3991 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
59 simpll 765 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → 𝑦 ∈ Fin)
60 elun1 4177 . . . . . . . . . . . . 13 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
6151simprd 494 . . . . . . . . . . . . 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 591 . . . . . . . . . . . 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 15738 . . . . . . . . . . 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 771 . . . . . . . . . . . 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 2892 . . . . . . . . . . . . . 14 𝑚𝐵
6665, 38, 45cbviun 5044 . . . . . . . . . . . . 13 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
6766fveq2i 6904 . . . . . . . . . . . 12 (vol*‘ 𝑘𝑦 𝐵) = (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
68 nfcv 2892 . . . . . . . . . . . . 13 𝑚(vol*‘𝐵)
6968, 42, 47cbvsumi 15701 . . . . . . . . . . . 12 Σ𝑘𝑦 (vol*‘𝐵) = Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵)
7064, 67, 693brtr3g 5186 . . . . . . . . . . 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 25503 . . . . . . . . . . 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 1368 . . . . . . . . . 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 4174 . . . . . . . . . . . . 13 {𝑧} ⊆ (𝑦 ∪ {𝑧})
74 vsnid 4670 . . . . . . . . . . . . 13 𝑧 ∈ {𝑧}
7573, 74sselii 3976 . . . . . . . . . . . 12 𝑧 ∈ (𝑦 ∪ {𝑧})
76 nfcsb1v 3917 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵
7776, 39nfss 3972 . . . . . . . . . . . . . 14 𝑘𝑧 / 𝑘𝐵 ⊆ ℝ
7841, 76nffv 6911 . . . . . . . . . . . . . . 15 𝑘(vol*‘𝑧 / 𝑘𝐵)
7978nfel1 2909 . . . . . . . . . . . . . 14 𝑘(vol*‘𝑧 / 𝑘𝐵) ∈ ℝ
8077, 79nfan 1895 . . . . . . . . . . . . 13 𝑘(𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3906 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281sseq1d 4011 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → (𝐵 ⊆ ℝ ↔ 𝑧 / 𝑘𝐵 ⊆ ℝ))
8381fveq2d 6905 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (vol*‘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
8483eleq1d 2811 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((vol*‘𝐵) ∈ ℝ ↔ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 630 . . . . . . . . . . . . 13 (𝑘 = 𝑧 → ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3596 . . . . . . . . . . . 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 494 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℝ)
8972, 88readdcld 11293 . . . . . . . . 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 5102 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
91 vex 3466 . . . . . . . . . . . . . 14 𝑧 ∈ V
92 csbeq1 3895 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
9391, 92iunxsn 5099 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
9493uneq2i 4160 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9590, 94eqtri 2754 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
9695fveq2i 6904 . . . . . . . . . 10 (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol*‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
97 ovolun 25519 . . . . . . . . . . 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 836 . . . . . . . . . 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, 98eqbrtrid 5188 . . . . . . . . 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 25503 . . . . . . . . 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 1368 . . . . . . . 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 9081 . . . . . . . . . . 11 {𝑧} ∈ Fin
103 unfi 9210 . . . . . . . . . . 11 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
104102, 103mpan2 689 . . . . . . . . . 10 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
105104ad2antrr 724 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) ∈ Fin)
106105, 61fsumrecl 15738 . . . . . . . 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 11878 . . . . . . . . 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 767 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → ¬ 𝑧𝑦)
109 disjsn 4720 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
110108, 109sylibr 233 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∩ {𝑧}) = ∅)
111 eqidd 2727 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
11261recnd 11292 . . . . . . . . . . 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 15745 . . . . . . . . . 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 11292 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ (vol*‘ 𝑘𝑦 𝐵) ≤ Σ𝑘𝑦 (vol*‘𝐵))) → (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ)
11592fveq2d 6905 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
116115sumsn 15750 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol*‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol*‘𝑚 / 𝑘𝐵) = (vol*‘𝑧 / 𝑘𝐵))
11791, 114, 116sylancr 585 . . . . . . . . . . 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 7440 . . . . . . . . . 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 2766 . . . . . . . . 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 5181 . . . . . . . 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 11421 . . . . . . 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 5044 . . . . . . . 8 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
123122fveq2i 6904 . . . . . . 7 (vol*‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol*‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
12468, 42, 47cbvsumi 15701 . . . . . . 7 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol*‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol*‘𝑚 / 𝑘𝐵)
125121, 123, 1243brtr4g 5187 . . . . . 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 419 . . . . 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 9203 . 2 (𝐴 ∈ Fin → (∀𝑘𝐴 (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (vol*‘ 𝑘𝐴 𝐵) ≤ Σ𝑘𝐴 (vol*‘𝐵)))
130129imp 405 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 394   = wceq 1534  wcel 2099  wral 3051  Vcvv 3462  csb 3892  cun 3945  cin 3946  wss 3947  c0 4325  {csn 4633   ciun 5001   class class class wbr 5153  cfv 6554  (class class class)co 7424  Fincfn 8974  cc 11156  cr 11157  0cc0 11158   + caddc 11161  cle 11299  Σcsu 15690  vol*covol 25482
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9684  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235  ax-pre-sup 11236
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-isom 6563  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-of 7690  df-om 7877  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-1o 8496  df-2o 8497  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-sup 9485  df-inf 9486  df-oi 9553  df-dju 9944  df-card 9982  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12611  df-uz 12875  df-q 12985  df-rp 13029  df-xadd 13147  df-ioo 13382  df-ico 13384  df-icc 13385  df-fz 13539  df-fzo 13682  df-fl 13812  df-seq 14022  df-exp 14082  df-hash 14348  df-cj 15104  df-re 15105  df-im 15106  df-sqrt 15240  df-abs 15241  df-clim 15490  df-sum 15691  df-xmet 21336  df-met 21337  df-ovol 25484
This theorem is referenced by:  volfiniun  25567  uniioombllem3a  25604  uniioombllem4  25606  i1fd  25701  i1fadd  25715  i1fmul  25716  volsupnfl  37366
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