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Theorem volfiniun 23122
 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 3115 . . . . 5 (𝑤 = ∅ → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2 disjeq1 4560 . . . . 5 (𝑤 = ∅ → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ ∅ 𝐵))
31, 2anbi12d 743 . . . 4 (𝑤 = ∅ → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵)))
4 iuneq1 4470 . . . . . 6 (𝑤 = ∅ → 𝑘𝑤 𝐵 = 𝑘 ∈ ∅ 𝐵)
54fveq2d 6107 . . . . 5 (𝑤 = ∅ → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ ∅ 𝐵))
6 sumeq1 14267 . . . . 5 (𝑤 = ∅ → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
75, 6eqeq12d 2625 . . . 4 (𝑤 = ∅ → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)))
83, 7imbi12d 333 . . 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 3115 . . . . 5 (𝑤 = 𝑦 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10 disjeq1 4560 . . . . 5 (𝑤 = 𝑦 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝑦 𝐵))
119, 10anbi12d 743 . . . 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 4470 . . . . . 6 (𝑤 = 𝑦 𝑘𝑤 𝐵 = 𝑘𝑦 𝐵)
1312fveq2d 6107 . . . . 5 (𝑤 = 𝑦 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝑦 𝐵))
14 sumeq1 14267 . . . . 5 (𝑤 = 𝑦 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝑦 (vol‘𝐵))
1513, 14eqeq12d 2625 . . . 4 (𝑤 = 𝑦 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)))
1611, 15imbi12d 333 . . 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 3115 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18 disjeq1 4560 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
1917, 18anbi12d 743 . . . 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 4470 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑘𝑤 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
2120fveq2d 6107 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22 sumeq1 14267 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
2321, 22eqeq12d 2625 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
2419, 23imbi12d 333 . . 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 3115 . . . . 5 (𝑤 = 𝐴 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26 disjeq1 4560 . . . . 5 (𝑤 = 𝐴 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝐴 𝐵))
2725, 26anbi12d 743 . . . 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 4470 . . . . . 6 (𝑤 = 𝐴 𝑘𝑤 𝐵 = 𝑘𝐴 𝐵)
2928fveq2d 6107 . . . . 5 (𝑤 = 𝐴 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝐴 𝐵))
30 sumeq1 14267 . . . . 5 (𝑤 = 𝐴 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝐴 (vol‘𝐵))
3129, 30eqeq12d 2625 . . . 4 (𝑤 = 𝐴 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
3227, 31imbi12d 333 . . 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 23114 . . . . . . 7 ∅ ∈ dom vol
34 mblvol 23105 . . . . . . 7 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
3533, 34ax-mp 5 . . . . . 6 (vol‘∅) = (vol*‘∅)
36 ovol0 23068 . . . . . 6 (vol*‘∅) = 0
3735, 36eqtri 2632 . . . . 5 (vol‘∅) = 0
38 0iun 4513 . . . . . 6 𝑘 ∈ ∅ 𝐵 = ∅
3938fveq2i 6106 . . . . 5 (vol‘ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40 sum0 14299 . . . . 5 Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
4137, 39, 403eqtr4i 2642 . . . 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 3738 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
44 ssralv 3629 . . . . . . 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 4559 . . . . . . 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 588 . . . . 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 61 . . . 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 6556 . . . . . . . 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 4541 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
52 vex 3176 . . . . . . . . . . . . . 14 𝑧 ∈ V
53 csbeq1 3502 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
5452, 53iunxsn 4539 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
5554uneq2i 3726 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5651, 55eqtri 2632 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5756fveq2i 6106 . . . . . . . . . 10 (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
58 nfcv 2751 . . . . . . . . . . . . 13 𝑚𝐵
59 nfcsb1v 3515 . . . . . . . . . . . . 13 𝑘𝑚 / 𝑘𝐵
60 csbeq1a 3508 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
6158, 59, 60cbviun 4493 . . . . . . . . . . . 12 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
62 simpll 786 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑦 ∈ Fin)
63 simprl 790 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → 𝐵 ∈ dom vol)
6564ralimi 2936 . . . . . . . . . . . . . . 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 3629 . . . . . . . . . . . . . 14 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom vol → ∀𝑘𝑦 𝐵 ∈ dom vol))
6843, 66, 67mpsyl 66 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑘𝑦 𝐵 ∈ dom vol)
69 finiunmbl 23119 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ∀𝑘𝑦 𝐵 ∈ dom vol) → 𝑘𝑦 𝐵 ∈ dom vol)
7062, 68, 69syl2anc 691 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑘𝑦 𝐵 ∈ dom vol)
7161, 70syl5eqelr 2693 . . . . . . . . . . 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 3739 . . . . . . . . . . . . . 14 {𝑧} ⊆ (𝑦 ∪ {𝑧})
73 vsnid 4156 . . . . . . . . . . . . . 14 𝑧 ∈ {𝑧}
7472, 73sselii 3565 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
75 nfcsb1v 3515 . . . . . . . . . . . . . . . 16 𝑘𝑧 / 𝑘𝐵
7675nfel1 2765 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵 ∈ dom vol
77 nfcv 2751 . . . . . . . . . . . . . . . . 17 𝑘vol
7877, 75nffv 6110 . . . . . . . . . . . . . . . 16 𝑘(vol‘𝑧 / 𝑘𝐵)
7978nfel1 2765 . . . . . . . . . . . . . . 15 𝑘(vol‘𝑧 / 𝑘𝐵) ∈ ℝ
8076, 79nfan 1816 . . . . . . . . . . . . . 14 𝑘(𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3508 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ 𝑧 / 𝑘𝐵 ∈ dom vol))
8381fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘𝑧 / 𝑘𝐵))
8483eleq1d 2672 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 743 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3276 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8774, 63, 86mpsyl 66 . . . . . . . . . . . 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 474 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑧 / 𝑘𝐵 ∈ dom vol)
89 simplr 788 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑧𝑦)
90 elin 3758 . . . . . . . . . . . . . 14 (𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) ↔ (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵))
91 eliun 4460 . . . . . . . . . . . . . . . 16 (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵 ↔ ∃𝑚𝑦 𝑤𝑚 / 𝑘𝐵)
92 simplrr 797 . . . . . . . . . . . . . . . . . . . . 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 2751 . . . . . . . . . . . . . . . . . . . . . 22 𝑛𝐵
94 nfcsb1v 3515 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑛 / 𝑘𝐵
95 csbeq1a 3508 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
9693, 94, 95cbvdisj 4563 . . . . . . . . . . . . . . . . . . . . 21 (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵)
9792, 96sylib 207 . . . . . . . . . . . . . . . . . . . 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 1060 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚𝑦)
99 elun1 3742 . . . . . . . . . . . . . . . . . . . . 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 1061 . . . . . . . . . . . . . . . . . . . 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 1062 . . . . . . . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚𝑛 / 𝑘𝐵 = 𝑚 / 𝑘𝐵)
105 csbeq1 3502 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧𝑛 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
106104, 105disji 4570 . . . . . . . . . . . . . . . . . . . 20 ((Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵 ∧ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) ∧ (𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 = 𝑧)
10797, 100, 101, 102, 103, 106syl122anc 1327 . . . . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . . . . 18 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑧𝑦)
1091083exp2 1277 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑚𝑦 → (𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦))))
110109rexlimdv 3012 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (∃𝑚𝑦 𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦)))
11191, 110syl5bi 231 . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . 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, 112syl5bi 231 . . . . . . . . . . . . 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 188 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
115114eq0rdv 3931 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) = ∅)
116 mblvol 23105 . . . . . . . . . . . . 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 1830 . . . . . . . . . . . . . . . . . . . . 21 𝑚(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)
11959nfel1 2765 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑚 / 𝑘𝐵 ∈ dom vol
12077, 59nffv 6110 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(vol‘𝑚 / 𝑘𝐵)
121120nfel1 2765 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(vol‘𝑚 / 𝑘𝐵) ∈ ℝ
122119, 121nfan 1816 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)
12360eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ 𝑚 / 𝑘𝐵 ∈ dom vol))
12460fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘𝑚 / 𝑘𝐵))
125124eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
126123, 125anbi12d 743 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)))
127118, 122, 126cbvral 3143 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
12863, 127sylib 207 . . . . . . . . . . . . . . . . . . 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 2916 . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . 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 23106 . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚𝑦) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
134133ralrimiva 2949 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
135 iunss 4497 . . . . . . . . . . . . . 14 ( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
136134, 135sylibr 223 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
137 mblvol 23105 . . . . . . . . . . . . . . . . . 18 (𝑚 / 𝑘𝐵 ∈ dom vol → (vol‘𝑚 / 𝑘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
138137eleq1d 2672 . . . . . . . . . . . . . . . . 17 (𝑚 / 𝑘𝐵 ∈ dom vol → ((vol‘𝑚 / 𝑘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
139138biimpa 500 . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . 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 14312 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
143131adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
144143, 139jca 553 . . . . . . . . . . . . . . . . 17 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
145144ralimi 2936 . . . . . . . . . . . . . . . 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 3629 . . . . . . . . . . . . . . 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 66 . . . . . . . . . . . . . 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 23076 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
15062, 148, 149syl2anc 691 . . . . . . . . . . . . 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 23058 . . . . . . . . . . . . 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 1318 . . . . . . . . . . . 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 2688 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
15487simprd 478 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
155 volun 23120 . . . . . . . . . . 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 1326 . . . . . . . . . 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, 156syl5eq 2656 . . . . . . . . 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 4192 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
15989, 158sylibr 223 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
160 eqidd 2611 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
161 snfi 7923 . . . . . . . . . . . 12 {𝑧} ∈ Fin
162 unfi 8112 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
16362, 161, 162sylancl 693 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
164129simprd 478 . . . . . . . . . . . 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 9947 . . . . . . . . . . 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 14318 . . . . . . . . . 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 9947 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℂ)
16853fveq2d 6107 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
169168sumsn 14319 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
17052, 167, 169sylancr 694 . . . . . . . . . . 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 6565 . . . . . . . . . 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 2644 . . . . . . . . 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 2625 . . . . . . . 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 235 . . . . . . 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 6106 . . . . . . . 8 (vol‘ 𝑘𝑦 𝐵) = (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
176 nfcv 2751 . . . . . . . . 9 𝑚(vol‘𝐵)
177176, 120, 124cbvsumi 14275 . . . . . . . 8 Σ𝑘𝑦 (vol‘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵)
178175, 177eqeq12i 2624 . . . . . . 7 ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) ↔ (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵))
17958, 59, 60cbviun 4493 . . . . . . . . 9 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
180179fveq2i 6106 . . . . . . . 8 (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
181176, 120, 124cbvsumi 14275 . . . . . . . 8 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵)
182180, 181eqeq12i 2624 . . . . . . 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 284 . . . . . 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 449 . . . . 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 33 . . 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 8086 . 2 (𝐴 ∈ Fin → ((∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
1881873impib 1254 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ⦋csb 3499   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ ciun 4455  Disj wdisj 4553   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  ℂcc 9813  ℝcr 9814  0cc0 9815   + caddc 9818   ≤ cle 9954  Σcsu 14264  vol*covol 23038  volcvol 23039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-xmet 19560  df-met 19561  df-ovol 23040  df-vol 23041 This theorem is referenced by:  uniioovol  23153  uniioombllem4  23160  itg1addlem1  23265  volfiniune  29620
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