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Theorem volfiniun 25296
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 3320 . . . . 5 (𝑤 = ∅ → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2 disjeq1 5119 . . . . 5 (𝑤 = ∅ → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ ∅ 𝐵))
31, 2anbi12d 629 . . . 4 (𝑤 = ∅ → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵)))
4 iuneq1 5012 . . . . . 6 (𝑤 = ∅ → 𝑘𝑤 𝐵 = 𝑘 ∈ ∅ 𝐵)
54fveq2d 6894 . . . . 5 (𝑤 = ∅ → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ ∅ 𝐵))
6 sumeq1 15639 . . . . 5 (𝑤 = ∅ → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
75, 6eqeq12d 2746 . . . 4 (𝑤 = ∅ → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)))
83, 7imbi12d 343 . . 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 3320 . . . . 5 (𝑤 = 𝑦 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10 disjeq1 5119 . . . . 5 (𝑤 = 𝑦 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝑦 𝐵))
119, 10anbi12d 629 . . . 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 5012 . . . . . 6 (𝑤 = 𝑦 𝑘𝑤 𝐵 = 𝑘𝑦 𝐵)
1312fveq2d 6894 . . . . 5 (𝑤 = 𝑦 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝑦 𝐵))
14 sumeq1 15639 . . . . 5 (𝑤 = 𝑦 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝑦 (vol‘𝐵))
1513, 14eqeq12d 2746 . . . 4 (𝑤 = 𝑦 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)))
1611, 15imbi12d 343 . . 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 3320 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18 disjeq1 5119 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
1917, 18anbi12d 629 . . . 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 5012 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑘𝑤 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
2120fveq2d 6894 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22 sumeq1 15639 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
2321, 22eqeq12d 2746 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
2419, 23imbi12d 343 . . 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 3320 . . . . 5 (𝑤 = 𝐴 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26 disjeq1 5119 . . . . 5 (𝑤 = 𝐴 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝐴 𝐵))
2725, 26anbi12d 629 . . . 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 5012 . . . . . 6 (𝑤 = 𝐴 𝑘𝑤 𝐵 = 𝑘𝐴 𝐵)
2928fveq2d 6894 . . . . 5 (𝑤 = 𝐴 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝐴 𝐵))
30 sumeq1 15639 . . . . 5 (𝑤 = 𝐴 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝐴 (vol‘𝐵))
3129, 30eqeq12d 2746 . . . 4 (𝑤 = 𝐴 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
3227, 31imbi12d 343 . . 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 25288 . . . . . . 7 ∅ ∈ dom vol
34 mblvol 25279 . . . . . . 7 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
3533, 34ax-mp 5 . . . . . 6 (vol‘∅) = (vol*‘∅)
36 ovol0 25242 . . . . . 6 (vol*‘∅) = 0
3735, 36eqtri 2758 . . . . 5 (vol‘∅) = 0
38 0iun 5065 . . . . . 6 𝑘 ∈ ∅ 𝐵 = ∅
3938fveq2i 6893 . . . . 5 (vol‘ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40 sum0 15671 . . . . 5 Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
4137, 39, 403eqtr4i 2768 . . . 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 4171 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
44 ssralv 4049 . . . . . . 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 5118 . . . . . . 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 611 . . . . 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 7418 . . . . . . . 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 5096 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
52 vex 3476 . . . . . . . . . . . . . 14 𝑧 ∈ V
53 csbeq1 3895 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
5452, 53iunxsn 5093 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
5554uneq2i 4159 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5651, 55eqtri 2758 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5756fveq2i 6893 . . . . . . . . . 10 (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
58 nfcv 2901 . . . . . . . . . . . . 13 𝑚𝐵
59 nfcsb1v 3917 . . . . . . . . . . . . 13 𝑘𝑚 / 𝑘𝐵
60 csbeq1a 3906 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
6158, 59, 60cbviun 5038 . . . . . . . . . . . 12 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
62 simpll 763 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑦 ∈ Fin)
63 simprl 767 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → 𝐵 ∈ dom vol)
6564ralimi 3081 . . . . . . . . . . . . . . 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 4049 . . . . . . . . . . . . . 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 25293 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ∀𝑘𝑦 𝐵 ∈ dom vol) → 𝑘𝑦 𝐵 ∈ dom vol)
7062, 68, 69syl2anc 582 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑘𝑦 𝐵 ∈ dom vol)
7161, 70eqeltrrid 2836 . . . . . . . . . . 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 4172 . . . . . . . . . . . . . 14 {𝑧} ⊆ (𝑦 ∪ {𝑧})
73 vsnid 4664 . . . . . . . . . . . . . 14 𝑧 ∈ {𝑧}
7472, 73sselii 3978 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
75 nfcsb1v 3917 . . . . . . . . . . . . . . . 16 𝑘𝑧 / 𝑘𝐵
7675nfel1 2917 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵 ∈ dom vol
77 nfcv 2901 . . . . . . . . . . . . . . . . 17 𝑘vol
7877, 75nffv 6900 . . . . . . . . . . . . . . . 16 𝑘(vol‘𝑧 / 𝑘𝐵)
7978nfel1 2917 . . . . . . . . . . . . . . 15 𝑘(vol‘𝑧 / 𝑘𝐵) ∈ ℝ
8076, 79nfan 1900 . . . . . . . . . . . . . 14 𝑘(𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3906 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281eleq1d 2816 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ 𝑧 / 𝑘𝐵 ∈ dom vol))
8381fveq2d 6894 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘𝑧 / 𝑘𝐵))
8483eleq1d 2816 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 629 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3599 . . . . . . . . . . . . 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 493 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑧 / 𝑘𝐵 ∈ dom vol)
89 simplr 765 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑧𝑦)
90 elin 3963 . . . . . . . . . . . . . 14 (𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) ↔ (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵))
91 eliun 5000 . . . . . . . . . . . . . . . 16 (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵 ↔ ∃𝑚𝑦 𝑤𝑚 / 𝑘𝐵)
92 simplrr 774 . . . . . . . . . . . . . . . . . . . . 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 2901 . . . . . . . . . . . . . . . . . . . . . 22 𝑛𝐵
94 nfcsb1v 3917 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑛 / 𝑘𝐵
95 csbeq1a 3906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
9693, 94, 95cbvdisj 5122 . . . . . . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚𝑦)
99 elun1 4175 . . . . . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . . . . . . 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 3895 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚𝑛 / 𝑘𝐵 = 𝑚 / 𝑘𝐵)
105 csbeq1 3895 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧𝑛 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
106104, 105disji 5130 . . . . . . . . . . . . . . . . . . . 20 ((Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵 ∧ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) ∧ (𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 = 𝑧)
10797, 100, 101, 102, 103, 106syl122anc 1377 . . . . . . . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . . . . . . 18 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑧𝑦)
1091083exp2 1352 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑚𝑦 → (𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦))))
110109rexlimdv 3151 . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . 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 4403 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) = ∅)
116 mblvol 25279 . . . . . . . . . . . . 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 1915 . . . . . . . . . . . . . . . . . . . . 21 𝑚(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)
11959nfel1 2917 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑚 / 𝑘𝐵 ∈ dom vol
12077, 59nffv 6900 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(vol‘𝑚 / 𝑘𝐵)
121120nfel1 2917 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(vol‘𝑚 / 𝑘𝐵) ∈ ℝ
122119, 121nfan 1900 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)
12360eleq1d 2816 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ 𝑚 / 𝑘𝐵 ∈ dom vol))
12460fveq2d 6894 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘𝑚 / 𝑘𝐵))
125124eleq1d 2816 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
126123, 125anbi12d 629 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)))
127118, 122, 126cbvralw 3301 . . . . . . . . . . . . . . . . . . . 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 3246 . . . . . . . . . . . . . . . . . 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 493 . . . . . . . . . . . . . . . . 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 25280 . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚𝑦) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
134133ralrimiva 3144 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
135 iunss 5047 . . . . . . . . . . . . . 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 25279 . . . . . . . . . . . . . . . . . 18 (𝑚 / 𝑘𝐵 ∈ dom vol → (vol‘𝑚 / 𝑘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
138137eleq1d 2816 . . . . . . . . . . . . . . . . 17 (𝑚 / 𝑘𝐵 ∈ dom vol → ((vol‘𝑚 / 𝑘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
139138biimpa 475 . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . 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 15684 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
143131adantr 479 . . . . . . . . . . . . . . . . . 18 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
144143, 139jca 510 . . . . . . . . . . . . . . . . 17 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
145144ralimi 3081 . . . . . . . . . . . . . . . 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 4049 . . . . . . . . . . . . . . 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 25250 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
15062, 148, 149syl2anc 582 . . . . . . . . . . . . 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 25232 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 2831 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
15487simprd 494 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
155 volun 25294 . . . . . . . . . . 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 1376 . . . . . . . . . 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 2782 . . . . . . . . 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 4714 . . . . . . . . . . . 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 2731 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
161 snfi 9046 . . . . . . . . . . . 12 {𝑧} ∈ Fin
162 unfi 9174 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
16362, 161, 162sylancl 584 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
164129simprd 494 . . . . . . . . . . . 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 11246 . . . . . . . . . . 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 15691 . . . . . . . . . 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 11246 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℂ)
16853fveq2d 6894 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
169168sumsn 15696 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
17052, 167, 169sylancr 585 . . . . . . . . . . 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 7427 . . . . . . . . . 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 2770 . . . . . . . . 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 2746 . . . . . . . 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, 173imbitrrid 245 . . . . . . 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 6893 . . . . . . . 8 (vol‘ 𝑘𝑦 𝐵) = (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
176 nfcv 2901 . . . . . . . . 9 𝑚(vol‘𝐵)
177176, 120, 124cbvsumi 15647 . . . . . . . 8 Σ𝑘𝑦 (vol‘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵)
178175, 177eqeq12i 2748 . . . . . . 7 ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) ↔ (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵))
17958, 59, 60cbviun 5038 . . . . . . . . 9 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
180179fveq2i 6893 . . . . . . . 8 (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
181176, 120, 124cbvsumi 15647 . . . . . . . 8 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵)
182180, 181eqeq12i 2748 . . . . . . 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 295 . . . . . 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 411 . . . . 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 9167 . 2 (𝐴 ∈ Fin → ((∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
1881873impib 1114 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 394  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  csb 3892  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627   ciun 4996  Disj wdisj 5112   class class class wbr 5147  dom cdm 5675  cfv 6542  (class class class)co 7411  Fincfn 8941  cc 11110  cr 11111  0cc0 11112   + caddc 11115  cle 11253  Σcsu 15636  vol*covol 25211  volcvol 25212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-xadd 13097  df-ioo 13332  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-xmet 21137  df-met 21138  df-ovol 25213  df-vol 25214
This theorem is referenced by:  uniioovol  25328  uniioombllem4  25335  itg1addlem1  25441  volfiniune  33526
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