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Theorem volfiniun 24155
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 3361 . . . . 5 (𝑤 = ∅ → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
2 disjeq1 5005 . . . . 5 (𝑤 = ∅ → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ ∅ 𝐵))
31, 2anbi12d 633 . . . 4 (𝑤 = ∅ → ((∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝑤 𝐵) ↔ (∀𝑘 ∈ ∅ (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ ∅ 𝐵)))
4 iuneq1 4900 . . . . . 6 (𝑤 = ∅ → 𝑘𝑤 𝐵 = 𝑘 ∈ ∅ 𝐵)
54fveq2d 6653 . . . . 5 (𝑤 = ∅ → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ ∅ 𝐵))
6 sumeq1 15041 . . . . 5 (𝑤 = ∅ → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵))
75, 6eqeq12d 2817 . . . 4 (𝑤 = ∅ → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ ∅ 𝐵) = Σ𝑘 ∈ ∅ (vol‘𝐵)))
83, 7imbi12d 348 . . 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 3361 . . . . 5 (𝑤 = 𝑦 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝑦 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
10 disjeq1 5005 . . . . 5 (𝑤 = 𝑦 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝑦 𝐵))
119, 10anbi12d 633 . . . 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 4900 . . . . . 6 (𝑤 = 𝑦 𝑘𝑤 𝐵 = 𝑘𝑦 𝐵)
1312fveq2d 6653 . . . . 5 (𝑤 = 𝑦 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝑦 𝐵))
14 sumeq1 15041 . . . . 5 (𝑤 = 𝑦 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝑦 (vol‘𝐵))
1513, 14eqeq12d 2817 . . . 4 (𝑤 = 𝑦 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵)))
1611, 15imbi12d 348 . . 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 3361 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
18 disjeq1 5005 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (Disj 𝑘𝑤 𝐵Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
1917, 18anbi12d 633 . . . 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 4900 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → 𝑘𝑤 𝐵 = 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)
2120fveq2d 6653 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵))
22 sumeq1 15041 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵))
2321, 22eqeq12d 2817 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵)))
2419, 23imbi12d 348 . . 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 3361 . . . . 5 (𝑤 = 𝐴 → (∀𝑘𝑤 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ)))
26 disjeq1 5005 . . . . 5 (𝑤 = 𝐴 → (Disj 𝑘𝑤 𝐵Disj 𝑘𝐴 𝐵))
2725, 26anbi12d 633 . . . 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 4900 . . . . . 6 (𝑤 = 𝐴 𝑘𝑤 𝐵 = 𝑘𝐴 𝐵)
2928fveq2d 6653 . . . . 5 (𝑤 = 𝐴 → (vol‘ 𝑘𝑤 𝐵) = (vol‘ 𝑘𝐴 𝐵))
30 sumeq1 15041 . . . . 5 (𝑤 = 𝐴 → Σ𝑘𝑤 (vol‘𝐵) = Σ𝑘𝐴 (vol‘𝐵))
3129, 30eqeq12d 2817 . . . 4 (𝑤 = 𝐴 → ((vol‘ 𝑘𝑤 𝐵) = Σ𝑘𝑤 (vol‘𝐵) ↔ (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
3227, 31imbi12d 348 . . 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 24147 . . . . . . 7 ∅ ∈ dom vol
34 mblvol 24138 . . . . . . 7 (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
3533, 34ax-mp 5 . . . . . 6 (vol‘∅) = (vol*‘∅)
36 ovol0 24101 . . . . . 6 (vol*‘∅) = 0
3735, 36eqtri 2824 . . . . 5 (vol‘∅) = 0
38 0iun 4952 . . . . . 6 𝑘 ∈ ∅ 𝐵 = ∅
3938fveq2i 6652 . . . . 5 (vol‘ 𝑘 ∈ ∅ 𝐵) = (vol‘∅)
40 sum0 15074 . . . . 5 Σ𝑘 ∈ ∅ (vol‘𝐵) = 0
4137, 39, 403eqtr4i 2834 . . . 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 4102 . . . . . . 7 𝑦 ⊆ (𝑦 ∪ {𝑧})
44 ssralv 3984 . . . . . . 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 5004 . . . . . . 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 615 . . . . 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 7146 . . . . . . . 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 4982 . . . . . . . . . . . 12 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵)
52 vex 3447 . . . . . . . . . . . . . 14 𝑧 ∈ V
53 csbeq1 3834 . . . . . . . . . . . . . 14 (𝑚 = 𝑧𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
5452, 53iunxsn 4979 . . . . . . . . . . . . 13 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵 = 𝑧 / 𝑘𝐵
5554uneq2i 4090 . . . . . . . . . . . 12 ( 𝑚𝑦 𝑚 / 𝑘𝐵 𝑚 ∈ {𝑧}𝑚 / 𝑘𝐵) = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5651, 55eqtri 2824 . . . . . . . . . . 11 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵 = ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵)
5756fveq2i 6652 . . . . . . . . . 10 (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵) = (vol‘( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
58 nfcv 2958 . . . . . . . . . . . . 13 𝑚𝐵
59 nfcsb1v 3855 . . . . . . . . . . . . 13 𝑘𝑚 / 𝑘𝐵
60 csbeq1a 3845 . . . . . . . . . . . . 13 (𝑘 = 𝑚𝐵 = 𝑚 / 𝑘𝐵)
6158, 59, 60cbviun 4926 . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) → 𝐵 ∈ dom vol)
6564ralimi 3131 . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . 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 24152 . . . . . . . . . . . . 13 ((𝑦 ∈ Fin ∧ ∀𝑘𝑦 𝐵 ∈ dom vol) → 𝑘𝑦 𝐵 ∈ dom vol)
7062, 68, 69syl2anc 587 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑘𝑦 𝐵 ∈ dom vol)
7161, 70eqeltrrid 2898 . . . . . . . . . . 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 4103 . . . . . . . . . . . . . 14 {𝑧} ⊆ (𝑦 ∪ {𝑧})
73 vsnid 4565 . . . . . . . . . . . . . 14 𝑧 ∈ {𝑧}
7472, 73sselii 3915 . . . . . . . . . . . . 13 𝑧 ∈ (𝑦 ∪ {𝑧})
75 nfcsb1v 3855 . . . . . . . . . . . . . . . 16 𝑘𝑧 / 𝑘𝐵
7675nfel1 2974 . . . . . . . . . . . . . . 15 𝑘𝑧 / 𝑘𝐵 ∈ dom vol
77 nfcv 2958 . . . . . . . . . . . . . . . . 17 𝑘vol
7877, 75nffv 6659 . . . . . . . . . . . . . . . 16 𝑘(vol‘𝑧 / 𝑘𝐵)
7978nfel1 2974 . . . . . . . . . . . . . . 15 𝑘(vol‘𝑧 / 𝑘𝐵) ∈ ℝ
8076, 79nfan 1900 . . . . . . . . . . . . . 14 𝑘(𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
81 csbeq1a 3845 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧𝐵 = 𝑧 / 𝑘𝐵)
8281eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → (𝐵 ∈ dom vol ↔ 𝑧 / 𝑘𝐵 ∈ dom vol))
8381fveq2d 6653 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑧 → (vol‘𝐵) = (vol‘𝑧 / 𝑘𝐵))
8483eleq1d 2877 . . . . . . . . . . . . . . 15 (𝑘 = 𝑧 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ))
8582, 84anbi12d 633 . . . . . . . . . . . . . 14 (𝑘 = 𝑧 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑧 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)))
8680, 85rspc 3562 . . . . . . . . . . . . 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 498 . . . . . . . . . . 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 3900 . . . . . . . . . . . . . 14 (𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) ↔ (𝑤 𝑚𝑦 𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵))
91 eliun 4888 . . . . . . . . . . . . . . . 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 2958 . . . . . . . . . . . . . . . . . . . . . 22 𝑛𝐵
94 nfcsb1v 3855 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑛 / 𝑘𝐵
95 csbeq1a 3845 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
9693, 94, 95cbvdisj 5008 . . . . . . . . . . . . . . . . . . . . 21 (Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵)
9792, 96sylib 221 . . . . . . . . . . . . . . . . . . . 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 1191 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚𝑦)
99 elun1 4106 . . . . . . . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . . . . 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 3834 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑚𝑛 / 𝑘𝐵 = 𝑚 / 𝑘𝐵)
105 csbeq1 3834 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧𝑛 / 𝑘𝐵 = 𝑧 / 𝑘𝐵)
106104, 105disji 5016 . . . . . . . . . . . . . . . . . . . 20 ((Disj 𝑛 ∈ (𝑦 ∪ {𝑧})𝑛 / 𝑘𝐵 ∧ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) ∧ (𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑚 = 𝑧)
10797, 100, 101, 102, 103, 106syl122anc 1376 . . . . . . . . . . . . . . . . . . 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 2894 . . . . . . . . . . . . . . . . . 18 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ (𝑚𝑦𝑤𝑚 / 𝑘𝐵𝑤𝑧 / 𝑘𝐵)) → 𝑧𝑦)
1091083exp2 1351 . . . . . . . . . . . . . . . . 17 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑚𝑦 → (𝑤𝑚 / 𝑘𝐵 → (𝑤𝑧 / 𝑘𝐵𝑧𝑦))))
110109rexlimdv 3245 . . . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . 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 245 . . . . . . . . . . . . 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 201 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ¬ 𝑤 ∈ ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵))
115114eq0rdv 4315 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ( 𝑚𝑦 𝑚 / 𝑘𝐵𝑧 / 𝑘𝐵) = ∅)
116 mblvol 24138 . . . . . . . . . . . . 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 2974 . . . . . . . . . . . . . . . . . . . . . 22 𝑘𝑚 / 𝑘𝐵 ∈ dom vol
12077, 59nffv 6659 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(vol‘𝑚 / 𝑘𝐵)
121120nfel1 2974 . . . . . . . . . . . . . . . . . . . . . 22 𝑘(vol‘𝑚 / 𝑘𝐵) ∈ ℝ
122119, 121nfan 1900 . . . . . . . . . . . . . . . . . . . . 21 𝑘(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)
12360eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → (𝐵 ∈ dom vol ↔ 𝑚 / 𝑘𝐵 ∈ dom vol))
12460fveq2d 6653 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑚 → (vol‘𝐵) = (vol‘𝑚 / 𝑘𝐵))
125124eleq1d 2877 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 𝑚 → ((vol‘𝐵) ∈ ℝ ↔ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
126123, 125anbi12d 633 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑚 → ((𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ (𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ)))
127118, 122, 126cbvralw 3390 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})(𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ))
12863, 127sylib 221 . . . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . . . 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 24139 . . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . . 15 ((((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) ∧ 𝑚𝑦) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
134133ralrimiva 3152 . . . . . . . . . . . . . 14 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
135 iunss 4935 . . . . . . . . . . . . . 14 ( 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ ↔ ∀𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
136134, 135sylibr 237 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → 𝑚𝑦 𝑚 / 𝑘𝐵 ⊆ ℝ)
137 mblvol 24138 . . . . . . . . . . . . . . . . . 18 (𝑚 / 𝑘𝐵 ∈ dom vol → (vol‘𝑚 / 𝑘𝐵) = (vol*‘𝑚 / 𝑘𝐵))
138137eleq1d 2877 . . . . . . . . . . . . . . . . 17 (𝑚 / 𝑘𝐵 ∈ dom vol → ((vol‘𝑚 / 𝑘𝐵) ∈ ℝ ↔ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
139138biimpa 480 . . . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . . 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 15087 . . . . . . . . . . . . 13 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)
143131adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → 𝑚 / 𝑘𝐵 ⊆ ℝ)
144143, 139jca 515 . . . . . . . . . . . . . . . . 17 ((𝑚 / 𝑘𝐵 ∈ dom vol ∧ (vol‘𝑚 / 𝑘𝐵) ∈ ℝ) → (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ))
145144ralimi 3131 . . . . . . . . . . . . . . . 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 3984 . . . . . . . . . . . . . . 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 24109 . . . . . . . . . . . . . 14 ((𝑦 ∈ Fin ∧ ∀𝑚𝑦 (𝑚 / 𝑘𝐵 ⊆ ℝ ∧ (vol*‘𝑚 / 𝑘𝐵) ∈ ℝ)) → (vol*‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ≤ Σ𝑚𝑦 (vol*‘𝑚 / 𝑘𝐵))
15062, 148, 149syl2anc 587 . . . . . . . . . . . . 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 24091 . . . . . . . . . . . . 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 1368 . . . . . . . . . . . 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 2893 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) ∈ ℝ)
15487simprd 499 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℝ)
155 volun 24153 . . . . . . . . . . 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 1375 . . . . . . . . . 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 2848 . . . . . . . . 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 4610 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
15989, 158sylibr 237 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∩ {𝑧}) = ∅)
160 eqidd 2802 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}))
161 snfi 8581 . . . . . . . . . . . 12 {𝑧} ∈ Fin
162 unfi 8773 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
16362, 161, 162sylancl 589 . . . . . . . . . . 11 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (𝑦 ∪ {𝑧}) ∈ Fin)
164129simprd 499 . . . . . . . . . . . 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 10662 . . . . . . . . . . 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 15093 . . . . . . . . . 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 10662 . . . . . . . . . . . 12 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (∀𝑘 ∈ (𝑦 ∪ {𝑧})(𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵)) → (vol‘𝑧 / 𝑘𝐵) ∈ ℂ)
16853fveq2d 6653 . . . . . . . . . . . . 13 (𝑚 = 𝑧 → (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
169168sumsn 15097 . . . . . . . . . . . 12 ((𝑧 ∈ V ∧ (vol‘𝑧 / 𝑘𝐵) ∈ ℂ) → Σ𝑚 ∈ {𝑧} (vol‘𝑚 / 𝑘𝐵) = (vol‘𝑧 / 𝑘𝐵))
17052, 167, 169sylancr 590 . . . . . . . . . . 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 7155 . . . . . . . . . 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 2836 . . . . . . . . 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 2817 . . . . . . . 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 249 . . . . . . 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 6652 . . . . . . . 8 (vol‘ 𝑘𝑦 𝐵) = (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵)
176 nfcv 2958 . . . . . . . . 9 𝑚(vol‘𝐵)
177176, 120, 124cbvsumi 15050 . . . . . . . 8 Σ𝑘𝑦 (vol‘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵)
178175, 177eqeq12i 2816 . . . . . . 7 ((vol‘ 𝑘𝑦 𝐵) = Σ𝑘𝑦 (vol‘𝐵) ↔ (vol‘ 𝑚𝑦 𝑚 / 𝑘𝐵) = Σ𝑚𝑦 (vol‘𝑚 / 𝑘𝐵))
17958, 59, 60cbviun 4926 . . . . . . . . 9 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵 = 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵
180179fveq2i 6652 . . . . . . . 8 (vol‘ 𝑘 ∈ (𝑦 ∪ {𝑧})𝐵) = (vol‘ 𝑚 ∈ (𝑦 ∪ {𝑧})𝑚 / 𝑘𝐵)
181176, 120, 124cbvsumi 15050 . . . . . . . 8 Σ𝑘 ∈ (𝑦 ∪ {𝑧})(vol‘𝐵) = Σ𝑚 ∈ (𝑦 ∪ {𝑧})(vol‘𝑚 / 𝑘𝐵)
182180, 181eqeq12i 2816 . . . . . . 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 299 . . . . . 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 416 . . . . 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 8747 . 2 (𝐴 ∈ Fin → ((∀𝑘𝐴 (𝐵 ∈ dom vol ∧ (vol‘𝐵) ∈ ℝ) ∧ Disj 𝑘𝐴 𝐵) → (vol‘ 𝑘𝐴 𝐵) = Σ𝑘𝐴 (vol‘𝐵)))
1881873impib 1113 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 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  csb 3831  cun 3882  cin 3883  wss 3884  c0 4246  {csn 4528   ciun 4884  Disj wdisj 4998   class class class wbr 5033  dom cdm 5523  cfv 6328  (class class class)co 7139  Fincfn 8496  cc 10528  cr 10529  0cc0 10530   + caddc 10533  cle 10669  Σcsu 15038  vol*covol 24070  volcvol 24071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607  ax-pre-sup 10608
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-disj 4999  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-div 11291  df-nn 11630  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12382  df-xadd 12500  df-ioo 12734  df-ico 12736  df-icc 12737  df-fz 12890  df-fzo 13033  df-fl 13161  df-seq 13369  df-exp 13430  df-hash 13691  df-cj 14454  df-re 14455  df-im 14456  df-sqrt 14590  df-abs 14591  df-clim 14841  df-sum 15039  df-xmet 20088  df-met 20089  df-ovol 24072  df-vol 24073
This theorem is referenced by:  uniioovol  24187  uniioombllem4  24194  itg1addlem1  24300  volfiniune  31603
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