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Theorem volfiniun 24934
Description: The volume of a disjoint finite union of measurable sets is the sum of the measures. (Contributed by Mario Carneiro, 25-Jun-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
volfiniun ((𝐴 ∈ Fin ∧ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))
Distinct variable group:   𝐴,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem volfiniun
Dummy variables 𝑚 𝑛 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3308 . . . . 5 (𝑤 = ∅ → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2 disjeq1 5081 . . . . 5 (𝑤 = ∅ → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ ∅ 𝐵))
31, 2anbi12d 632 . . . 4 (𝑤 = ∅ → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵)))
4 iuneq1 4974 . . . . . 6 (𝑤 = ∅ → 𝑘𝑤 𝐵 = 𝑘 ∈ ∅ 𝐵)
54fveq2d 6850 . . . . 5 (𝑤 = ∅ → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ ∅ 𝐵))
6 sumeq1 15582 . . . . 5 (𝑤 = ∅ → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
75, 6eqeq12d 2749 . . . 4 (𝑤 = ∅ → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)))
83, 7imbi12d 345 . . 3 (𝑤 = ∅ → (((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) → (vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵)) ↔ ((∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵) → (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))))
9 raleq 3308 . . . . 5 (𝑤 = 𝑦 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10 disjeq1 5081 . . . . 5 (𝑤 = 𝑦 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝑦 𝐵))
119, 10anbi12d 632 . . . 4 (𝑤 = 𝑦 → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑦 𝐵)))
12 iuneq1 4974 . . . . . 6 (𝑤 = 𝑦 𝑘𝑤 𝐵 = 𝑘𝑦 𝐵)
1312fveq2d 6850 . . . . 5 (𝑤 = 𝑦 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝑦 𝐵))
14 sumeq1 15582 . . . . 5 (𝑤 = 𝑦 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝑦 (vol‘𝐵))
1513, 14eqeq12d 2749 . . . 4 (𝑤 = 𝑦 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)))
1611, 15imbi12d 345 . . 3 (𝑤 = 𝑦 → (((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) → (vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵)) ↔ ((∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑦 𝐵) → (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵))))
17 raleq 3308 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18 disjeq1 5081 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
1917, 18anbi12d 632 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)))
20 iuneq1 4974 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑘𝑤 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
2120fveq2d 6850 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22 sumeq1 15582 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
2321, 22eqeq12d 2749 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
2419, 23imbi12d 345 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → (((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) → (vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵)) ↔ ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
25 raleq 3308 . . . . 5 (𝑤 = 𝐴 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26 disjeq1 5081 . . . . 5 (𝑤 = 𝐴 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝐴 𝐵))
2725, 26anbi12d 632 . . . 4 (𝑤 = 𝐴 → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵)))
28 iuneq1 4974 . . . . . 6 (𝑤 = 𝐴 𝑘𝑤 𝐵 = 𝑘𝐴 𝐵)
2928fveq2d 6850 . . . . 5 (𝑤 = 𝐴 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝐴 𝐵))
30 sumeq1 15582 . . . . 5 (𝑤 = 𝐴 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝐴 (vol‘𝐵))
3129, 30eqeq12d 2749 . . . 4 (𝑤 = 𝐴 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
3227, 31imbi12d 345 . . 3 (𝑤 = 𝐴 → (((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) → (vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵)) ↔ ((∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))))
33 0mbl 24926 . . . . . . 7 ∅ ∈ dom vol
34 mblvol 24917 . . . . . . 7 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
3533, 34ax-mp 5 . . . . . 6 (vol‘∅) = (vol*‘∅)
36 ovol0 24880 . . . . . 6 (vol*‘∅) = 0
3735, 36eqtri 2761 . . . . 5 (vol‘∅) = 0
38 0iun 5027 . . . . . 6 𝑘 ∈ ∅ 𝐵 = ∅
3938fveq2i 6849 . . . . 5 (vol‘ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40 sum0 15614 . . . . 5 Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
4137, 39, 403eqtr4i 2771 . . . 4 (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)
4241a1i 11 . . 3 ((∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵) → (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
43 ssun1 4136 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
44 ssralv 4014 . . . . . . 7 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
4543, 44ax-mp 5 . . . . . 6 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ))
46 disjss1 5080 . . . . . . 7 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵Disj 𝑘𝑦 𝐵))
4743, 46ax-mp 5 . . . . . 6 (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵Disj 𝑘𝑦 𝐵)
4845, 47anim12i 614 . . . . 5 ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑦 𝐵))
4948imim1i 63 . . . 4 (((∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑦 𝐵) → (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)))
50 oveq1 7368 . . . . . . . 8 ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) → ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)) = (Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)))
51 iunxun 5058 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
52 vex 3451 . . . . . . . . . . . . . 14 𝑧 ∈ V
53 csbeq1 3862 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
5452, 53iunxsn 5055 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
5554uneq2i 4124 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5651, 55eqtri 2761 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5756fveq2i 6849 . . . . . . . . . 10 (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
58 nfcv 2904 . . . . . . . . . . . . 13 𝑚𝐵
59 nfcsb1v 3884 . . . . . . . . . . . . 13 𝑘𝑚 / 𝑘𝐵
60 csbeq1a 3873 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
6158, 59, 60cbviun 5000 . . . . . . . . . . . 12 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
62 simpll 766 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑦 ∈ Fin)
63 simprl 770 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ))
64 simpl 484 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → 𝐵 ∈ dom vol)
6564ralimi 3083 . . . . . . . . . . . . . . 15 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol)
6663, 65syl 17 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol)
67 ssralv 4014 . . . . . . . . . . . . . 14 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol → ∀𝑘𝑦 𝐵 ∈ dom vol))
6843, 66, 67mpsyl 68 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘𝑦 𝐵 ∈ dom vol)
69 finiunmbl 24931 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ∀𝑘𝑦 𝐵 ∈ dom vol) → 𝑘𝑦 𝐵 ∈ dom vol)
7062, 68, 69syl2anc 585 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑘𝑦 𝐵 ∈ dom vol)
7161, 70eqeltrrid 2839 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑚𝑦 𝑚 / 𝑘𝐵 ∈ dom vol)
72 ssun2 4137 . . . . . . . . . . . . . 14 {𝑧} ⊆ (𝑦 ∪ {𝑧})
73 vsnid 4627 . . . . . . . . . . . . . 14 𝑧 ∈ {𝑧}
7472, 73sselii 3945 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
75 nfcsb1v 3884 . . . . . . . . . . . . . . . 16 𝑘𝑧 / 𝑘𝐵
7675nfel1 2920 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵 ∈ dom vol
77 nfcv 2904 . . . . . . . . . . . . . . . . 17 𝑘vol
7877, 75nffv 6856 . . . . . . . . . . . . . . . 16 𝑘(vol‘𝑧 / 𝑘𝐵)
7978nfel1 2920 . . . . . . . . . . . . . . 15 𝑘(vol‘𝑧 / 𝑘𝐵) ∈ ℝ
8076, 79nfan 1903 . . . . . . . . . . . . . 14 𝑘(𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3873 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281eleq1d 2819 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ 𝑧 / 𝑘𝐵 ∈ dom vol))
8381fveq2d 6850 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘𝑧 / 𝑘𝐵))
8483eleq1d 2819 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 632 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3571 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8774, 63, 86mpsyl 68 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ))
8887simpld 496 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑧 / 𝑘𝐵 ∈ dom vol)
89 simplr 768 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑧𝑦)
90 elin 3930 . . . . . . . . . . . . . 14 (𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) ↔ (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵))
91 eliun 4962 . . . . . . . . . . . . . . . 16 (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵 ↔ ∃𝑚𝑦 𝑤𝑚 / 𝑘𝐵)
92 simplrr 777 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
93 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . 22 𝑛𝐵
94 nfcsb1v 3884 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑛 / 𝑘𝐵
95 csbeq1a 3873 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
9693, 94, 95cbvdisj 5084 . . . . . . . . . . . . . . . . . . . . 21 (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵)
9792, 96sylib 217 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵)
98 simpr1 1195 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚𝑦)
99 elun1 4140 . . . . . . . . . . . . . . . . . . . . 21 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
10098, 99syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 ∈ (𝑦 ∪ {𝑧}))
10174a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑧 ∈ (𝑦 ∪ {𝑧}))
102 simpr2 1196 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑤𝑚 / 𝑘𝐵)
103 simpr3 1197 . . . . . . . . . . . . . . . . . . . 20 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑤𝑧 / 𝑘𝐵)
104 csbeq1 3862 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚𝑛 / 𝑘𝐵 = 𝑚 / 𝑘𝐵)
105 csbeq1 3862 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧𝑛 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
106104, 105disji 5092 . . . . . . . . . . . . . . . . . . . 20 ((Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵 ∧ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) ∧ (𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 = 𝑧)
10797, 100, 101, 102, 103, 106syl122anc 1380 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 = 𝑧)
108107, 98eqeltrrd 2835 . . . . . . . . . . . . . . . . . 18 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑧𝑦)
1091083exp2 1355 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑚𝑦 → (𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦))))
110109rexlimdv 3147 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (∃𝑚𝑦 𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦)))
11191, 110biimtrid 241 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦)))
112111impd 412 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((𝑤 𝑚𝑦 𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵) → 𝑧𝑦))
11390, 112biimtrid 241 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) → 𝑧𝑦))
11489, 113mtod 197 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
115114eq0rdv 4368 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) = ∅)
116 mblvol 24917 . . . . . . . . . . . . 13 ( 𝑚𝑦 𝑚 / 𝑘𝐵 ∈ dom vol → (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵))
11771, 116syl 17 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵))
118 nfv 1918 . . . . . . . . . . . . . . . . . . . . 21 𝑚(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)
11959nfel1 2920 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑚 / 𝑘𝐵 ∈ dom vol
12077, 59nffv 6856 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(vol‘𝑚 / 𝑘𝐵)
121120nfel1 2920 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(vol‘𝑚 / 𝑘𝐵) ∈ ℝ
122119, 121nfan 1903 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)
12360eleq1d 2819 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ 𝑚 / 𝑘𝐵 ∈ dom vol))
12460fveq2d 6850 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘𝑚 / 𝑘𝐵))
125124eleq1d 2819 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
126123, 125anbi12d 632 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)))
127118, 122, 126cbvralw 3288 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
12863, 127sylib 217 . . . . . . . . . . . . . . . . . . 19 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
129128r19.21bi 3233 . . . . . . . . . . . . . . . . . 18 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
130129simpld 496 . . . . . . . . . . . . . . . . 17 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → 𝑚 / 𝑘𝐵 ∈ dom vol)
131 mblss 24918 . . . . . . . . . . . . . . . . 17 (𝑚 / 𝑘𝐵 ∈ dom vol → 𝑚 / 𝑘𝐵 ⊆ ℝ)
132130, 131syl 17 . . . . . . . . . . . . . . . 16 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
13399, 132sylan2 594 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚𝑦) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
134133ralrimiva 3140 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
135 iunss 5009 . . . . . . . . . . . . . 14 ( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
136134, 135sylibr 233 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
137 mblvol 24917 . . . . . . . . . . . . . . . . . 18 (𝑚 / 𝑘𝐵 ∈ dom vol → (vol‘𝑚 / 𝑘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
138137eleq1d 2819 . . . . . . . . . . . . . . . . 17 (𝑚 / 𝑘𝐵 ∈ dom vol → ((vol‘𝑚 / 𝑘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
139138biimpa 478 . . . . . . . . . . . . . . . 16 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
140129, 139syl 17 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
14199, 140sylan2 594 . . . . . . . . . . . . . 14 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚𝑦) → (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
14262, 141fsumrecl 15627 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
143131adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
144143, 139jca 513 . . . . . . . . . . . . . . . . 17 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
145144ralimi 3083 . . . . . . . . . . . . . . . 16 (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
146128, 145syl 17 . . . . . . . . . . . . . . 15 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
147 ssralv 4014 . . . . . . . . . . . . . . 15 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ) → ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)))
14843, 146, 147mpsyl 68 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
149 ovolfiniun 24888 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
15062, 148, 149syl2anc 585 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
151 ovollecl 24870 . . . . . . . . . . . . 13 (( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ∧ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ ∧ (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
152136, 142, 150, 151syl3anc 1372 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
153117, 152eqeltrd 2834 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
15487simprd 497 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
155 volun 24932 . . . . . . . . . . 11 ((( 𝑚𝑦 𝑚 / 𝑘𝐵 ∈ dom vol ∧ 𝑧 / 𝑘𝐵 ∈ dom vol ∧ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) = ∅) ∧ ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)) → (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)) = ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)))
15671, 88, 115, 153, 154, 155syl32anc 1379 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)) = ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)))
15757, 156eqtrid 2785 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)))
158 disjsn 4676 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
15989, 158sylibr 233 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
160 eqidd 2734 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
161 snfi 8994 . . . . . . . . . . . 12 {𝑧} ∈ Fin
162 unfi 9122 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
16362, 161, 162sylancl 587 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
164129simprd 497 . . . . . . . . . . . 12 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)
165164recnd 11191 . . . . . . . . . . 11 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚 ∈ (𝑦 ∪ {𝑧})) → (vol‘𝑚 / 𝑘𝐵) ∈ ℂ)
166159, 160, 163, 165fsumsplit 15634 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵) = (Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) + Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵)))
167154recnd 11191 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℂ)
16853fveq2d 6850 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
169168sumsn 15639 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
17052, 167, 169sylancr 588 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
171170oveq2d 7377 . . . . . . . . . 10 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) + Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵)) = (Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)))
172166, 171eqtrd 2773 . . . . . . . . 9 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵) = (Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)))
173157, 172eqeq12d 2749 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵) ↔ ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵)) = (Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) + (vol‘𝑧 / 𝑘𝐵))))
17450, 173syl5ibr 246 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵) → (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵)))
17561fveq2i 6849 . . . . . . . 8 (vol‘ 𝑘𝑦 𝐵) = (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
176 nfcv 2904 . . . . . . . . 9 𝑚(vol‘𝐵)
177176, 120, 124cbvsumi 15590 . . . . . . . 8 Σ𝑘𝑦 (vol‘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵)
178175, 177eqeq12i 2751 . . . . . . 7 ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) ↔ (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵))
17958, 59, 60cbviun 5000 . . . . . . . . 9 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
180179fveq2i 6849 . . . . . . . 8 (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
181176, 120, 124cbvsumi 15590 . . . . . . . 8 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵)
182180, 181eqeq12i 2751 . . . . . . 7 ((vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) ↔ (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵))
183174, 178, 1823imtr4g 296 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) → (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
184183ex 414 . . . . 5 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) → (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
185184a2d 29 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
18649, 185syl5 34 . . 3 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (((∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑦 𝐵) → (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)) → ((∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) → (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))))
1878, 16, 24, 32, 42, 186findcard2s 9115 . 2 (𝐴 ∈ Fin → ((∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
1881873impib 1117 1 ((𝐴 ∈ Fin ∧ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070  Vcvv 3447  csb 3859  cun 3912  cin 3913  wss 3914  c0 4286  {csn 4590   ciun 4958  Disj wdisj 5074   class class class wbr 5109  dom cdm 5637  cfv 6500  (class class class)co 7361  Fincfn 8889  cc 11057  cr 11058  0cc0 11059   + caddc 11062  cle 11198  Σcsu 15579  vol*covol 24849  volcvol 24850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-disj 5075  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-oi 9454  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-xadd 13042  df-ioo 13277  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-fl 13706  df-seq 13916  df-exp 13977  df-hash 14240  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-clim 15379  df-sum 15580  df-xmet 20812  df-met 20813  df-ovol 24851  df-vol 24852
This theorem is referenced by:  uniioovol  24966  uniioombllem4  24973  itg1addlem1  25079  volfiniune  32893
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