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Theorem volfiniun 23361
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 3168 . . . . 5 (𝑤 = ∅ → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2 disjeq1 4659 . . . . 5 (𝑤 = ∅ → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ ∅ 𝐵))
31, 2anbi12d 747 . . . 4 (𝑤 = ∅ → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵)))
4 iuneq1 4566 . . . . . 6 (𝑤 = ∅ → 𝑘𝑤 𝐵 = 𝑘 ∈ ∅ 𝐵)
54fveq2d 6233 . . . . 5 (𝑤 = ∅ → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ ∅ 𝐵))
6 sumeq1 14463 . . . . 5 (𝑤 = ∅ → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
75, 6eqeq12d 2666 . . . 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 3168 . . . . 5 (𝑤 = 𝑦 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10 disjeq1 4659 . . . . 5 (𝑤 = 𝑦 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝑦 𝐵))
119, 10anbi12d 747 . . . 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 4566 . . . . . 6 (𝑤 = 𝑦 𝑘𝑤 𝐵 = 𝑘𝑦 𝐵)
1312fveq2d 6233 . . . . 5 (𝑤 = 𝑦 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝑦 𝐵))
14 sumeq1 14463 . . . . 5 (𝑤 = 𝑦 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝑦 (vol‘𝐵))
1513, 14eqeq12d 2666 . . . 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 3168 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18 disjeq1 4659 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
1917, 18anbi12d 747 . . . 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 4566 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑘𝑤 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
2120fveq2d 6233 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22 sumeq1 14463 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
2321, 22eqeq12d 2666 . . . 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 3168 . . . . 5 (𝑤 = 𝐴 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26 disjeq1 4659 . . . . 5 (𝑤 = 𝐴 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝐴 𝐵))
2725, 26anbi12d 747 . . . 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 4566 . . . . . 6 (𝑤 = 𝐴 𝑘𝑤 𝐵 = 𝑘𝐴 𝐵)
2928fveq2d 6233 . . . . 5 (𝑤 = 𝐴 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝐴 𝐵))
30 sumeq1 14463 . . . . 5 (𝑤 = 𝐴 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝐴 (vol‘𝐵))
3129, 30eqeq12d 2666 . . . 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 23353 . . . . . . 7 ∅ ∈ dom vol
34 mblvol 23344 . . . . . . 7 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
3533, 34ax-mp 5 . . . . . 6 (vol‘∅) = (vol*‘∅)
36 ovol0 23307 . . . . . 6 (vol*‘∅) = 0
3735, 36eqtri 2673 . . . . 5 (vol‘∅) = 0
38 0iun 4609 . . . . . 6 𝑘 ∈ ∅ 𝐵 = ∅
3938fveq2i 6232 . . . . 5 (vol‘ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40 sum0 14496 . . . . 5 Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
4137, 39, 403eqtr4i 2683 . . . 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 3809 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
44 ssralv 3699 . . . . . . 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 4658 . . . . . . 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 589 . . . . 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 6697 . . . . . . . 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 4637 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
52 vex 3234 . . . . . . . . . . . . . 14 𝑧 ∈ V
53 csbeq1 3569 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
5452, 53iunxsn 4635 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
5554uneq2i 3797 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5651, 55eqtri 2673 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5756fveq2i 6232 . . . . . . . . . 10 (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
58 nfcv 2793 . . . . . . . . . . . . 13 𝑚𝐵
59 nfcsb1v 3582 . . . . . . . . . . . . 13 𝑘𝑚 / 𝑘𝐵
60 csbeq1a 3575 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
6158, 59, 60cbviun 4589 . . . . . . . . . . . 12 𝑘𝑦 𝐵 = 𝑚𝑦 𝑚 / 𝑘𝐵
62 simpll 805 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑦 ∈ Fin)
63 simprl 809 . . . . . . . . . . . . . . 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 2981 . . . . . . . . . . . . . . 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 3699 . . . . . . . . . . . . . 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 23358 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ∀𝑘𝑦 𝐵 ∈ dom vol) → 𝑘𝑦 𝐵 ∈ dom vol)
7062, 68, 69syl2anc 694 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑘𝑦 𝐵 ∈ dom vol)
7161, 70syl5eqelr 2735 . . . . . . . . . . 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 3810 . . . . . . . . . . . . . 14 {𝑧} ⊆ (𝑦 ∪ {𝑧})
73 vsnid 4242 . . . . . . . . . . . . . 14 𝑧 ∈ {𝑧}
7472, 73sselii 3633 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
75 nfcsb1v 3582 . . . . . . . . . . . . . . . 16 𝑘𝑧 / 𝑘𝐵
7675nfel1 2808 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵 ∈ dom vol
77 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑘vol
7877, 75nffv 6236 . . . . . . . . . . . . . . . 16 𝑘(vol‘𝑧 / 𝑘𝐵)
7978nfel1 2808 . . . . . . . . . . . . . . 15 𝑘(vol‘𝑧 / 𝑘𝐵) ∈ ℝ
8076, 79nfan 1868 . . . . . . . . . . . . . 14 𝑘(𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3575 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ 𝑧 / 𝑘𝐵 ∈ dom vol))
8381fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘𝑧 / 𝑘𝐵))
8483eleq1d 2715 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 747 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3334 . . . . . . . . . . . . 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 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 807 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑧𝑦)
90 elin 3829 . . . . . . . . . . . . . 14 (𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) ↔ (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵))
91 eliun 4556 . . . . . . . . . . . . . . . 16 (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵 ↔ ∃𝑚𝑦 𝑤𝑚 / 𝑘𝐵)
92 simplrr 818 . . . . . . . . . . . . . . . . . . . . 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 2793 . . . . . . . . . . . . . . . . . . . . . 22 𝑛𝐵
94 nfcsb1v 3582 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑛 / 𝑘𝐵
95 csbeq1a 3575 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
9693, 94, 95cbvdisj 4662 . . . . . . . . . . . . . . . . . . . . 21 (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵)
9792, 96sylib 208 . . . . . . . . . . . . . . . . . . . 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 1087 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚𝑦)
99 elun1 3813 . . . . . . . . . . . . . . . . . . . . 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 1088 . . . . . . . . . . . . . . . . . . . 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 1089 . . . . . . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚𝑛 / 𝑘𝐵 = 𝑚 / 𝑘𝐵)
105 csbeq1 3569 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧𝑛 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
106104, 105disji 4669 . . . . . . . . . . . . . . . . . . . 20 ((Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵 ∧ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) ∧ (𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 = 𝑧)
10797, 100, 101, 102, 103, 106syl122anc 1375 . . . . . . . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . . . . . 18 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑧𝑦)
1091083exp2 1307 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑚𝑦 → (𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦))))
110109rexlimdv 3059 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . 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 189 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
115114eq0rdv 4012 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) = ∅)
116 mblvol 23344 . . . . . . . . . . . . 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 1883 . . . . . . . . . . . . . . . . . . . . 21 𝑚(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)
11959nfel1 2808 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑚 / 𝑘𝐵 ∈ dom vol
12077, 59nffv 6236 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(vol‘𝑚 / 𝑘𝐵)
121120nfel1 2808 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(vol‘𝑚 / 𝑘𝐵) ∈ ℝ
122119, 121nfan 1868 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)
12360eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ 𝑚 / 𝑘𝐵 ∈ dom vol))
12460fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘𝑚 / 𝑘𝐵))
125124eleq1d 2715 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
126123, 125anbi12d 747 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)))
127118, 122, 126cbvral 3197 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
12863, 127sylib 208 . . . . . . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . . . . . . 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 23345 . . . . . . . . . . . . . . . . 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 2995 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
135 iunss 4593 . . . . . . . . . . . . . 14 ( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
136134, 135sylibr 224 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
137 mblvol 23344 . . . . . . . . . . . . . . . . . 18 (𝑚 / 𝑘𝐵 ∈ dom vol → (vol‘𝑚 / 𝑘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
138137eleq1d 2715 . . . . . . . . . . . . . . . . 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 14509 . . . . . . . . . . . . 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 2981 . . . . . . . . . . . . . . . 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 3699 . . . . . . . . . . . . . . 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 23315 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
15062, 148, 149syl2anc 694 . . . . . . . . . . . . 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 23297 . . . . . . . . . . . . 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 1366 . . . . . . . . . . . 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 2730 . . . . . . . . . . 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 23359 . . . . . . . . . . 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 1374 . . . . . . . . . 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 2697 . . . . . . . . 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 4278 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
15989, 158sylibr 224 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
160 eqidd 2652 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
161 snfi 8079 . . . . . . . . . . . 12 {𝑧} ∈ Fin
162 unfi 8268 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
16362, 161, 162sylancl 695 . . . . . . . . . . 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 10106 . . . . . . . . . . 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 14515 . . . . . . . . . 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 10106 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℂ)
16853fveq2d 6233 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
169168sumsn 14519 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
17052, 167, 169sylancr 696 . . . . . . . . . . 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 6706 . . . . . . . . . 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 2685 . . . . . . . . 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 2666 . . . . . . . 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 236 . . . . . . 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 6232 . . . . . . . 8 (vol‘ 𝑘𝑦 𝐵) = (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
176 nfcv 2793 . . . . . . . . 9 𝑚(vol‘𝐵)
177176, 120, 124cbvsumi 14471 . . . . . . . 8 Σ𝑘𝑦 (vol‘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵)
178175, 177eqeq12i 2665 . . . . . . 7 ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) ↔ (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵))
17958, 59, 60cbviun 4589 . . . . . . . . 9 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
180179fveq2i 6232 . . . . . . . 8 (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
181176, 120, 124cbvsumi 14471 . . . . . . . 8 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵)
182180, 181eqeq12i 2665 . . . . . . 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 285 . . . . . 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 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 8242 . 2 (𝐴 ∈ Fin → ((∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
1881873impib 1281 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 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  csb 3566  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   ciun 4552  Disj wdisj 4652   class class class wbr 4685  dom cdm 5143  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  cr 9973  0cc0 9974   + caddc 9977  cle 10113  Σcsu 14460  vol*covol 23277  volcvol 23278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280
This theorem is referenced by:  uniioovol  23393  uniioombllem4  23400  itg1addlem1  23504  volfiniune  30421
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