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Theorem voliune 31481
Description: The Lebesgue measure function is countably additive. This formulation on the extended reals, allows for +∞ for the measure of any set in the sum. Cf. ovoliun 24098 and voliun 24147. (Contributed by Thierry Arnoux, 16-Oct-2017.)
Assertion
Ref Expression
voliune ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = Σ*𝑛 ∈ ℕ(vol‘𝐴))

Proof of Theorem voliune
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 r19.26 3168 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2 eqid 2819 . . . . . 6 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3 eqid 2819 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
42, 3voliun 24147 . . . . 5 ((∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
51, 4sylanbr 584 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
65an32s 650 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ 𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
7 nfra1 3217 . . . . . . 7 𝑛𝑛 ∈ ℕ 𝐴 ∈ dom vol
8 nfra1 3217 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
97, 8nfan 1894 . . . . . 6 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10 simpr 487 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11 rspa 3204 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12 volf 24122 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
1312ffvelrni 6843 . . . . . . . . . . 11 (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
1411, 13syl 17 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
153fvmpt2 6772 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1610, 14, 15syl2anc 586 . . . . . . . . 9 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1716adantlr 713 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1817ex 415 . . . . . . 7 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
199, 18ralrimi 3214 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
209, 19esumeq2d 31289 . . . . 5 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ*𝑛 ∈ ℕ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = Σ*𝑛 ∈ ℕ(vol‘𝐴))
21 simpr 487 . . . . . . . . 9 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
2221r19.21bi 3206 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
2314adantlr 713 . . . . . . . . 9 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24 0xr 10680 . . . . . . . . . . 11 0 ∈ ℝ*
25 pnfxr 10687 . . . . . . . . . . 11 +∞ ∈ ℝ*
26 elicc1 12774 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
2724, 25, 26mp2an 690 . . . . . . . . . 10 ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞))
2827simp2bi 1141 . . . . . . . . 9 ((vol‘𝐴) ∈ (0[,]+∞) → 0 ≤ (vol‘𝐴))
2923, 28syl 17 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘𝐴))
30 ltpnf 12507 . . . . . . . . 9 ((vol‘𝐴) ∈ ℝ → (vol‘𝐴) < +∞)
3122, 30syl 17 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) < +∞)
32 0re 10635 . . . . . . . . 9 0 ∈ ℝ
33 elico2 12792 . . . . . . . . 9 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
3432, 25, 33mp2an 690 . . . . . . . 8 ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
3522, 29, 31, 34syl3anbrc 1338 . . . . . . 7 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
369, 35, 3fmptdf 6874 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37 nfmpt1 5155 . . . . . . 7 𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
3837esumfsupre 31323 . . . . . 6 ((𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞) → Σ*𝑛 ∈ ℕ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
3936, 38syl 17 . . . . 5 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ*𝑛 ∈ ℕ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
4020, 39eqtr3d 2856 . . . 4 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ*𝑛 ∈ ℕ(vol‘𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
4140adantlr 713 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → Σ*𝑛 ∈ ℕ(vol‘𝐴) = sup(ran seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))), ℝ*, < ))
426, 41eqtr4d 2857 . 2 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ 𝐴) = Σ*𝑛 ∈ ℕ(vol‘𝐴))
43 simpr 487 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44 nfv 1909 . . . . . . . . 9 𝑘(vol‘𝐴) = +∞
45 nfcv 2975 . . . . . . . . . . 11 𝑛vol
46 nfcsb1v 3905 . . . . . . . . . . 11 𝑛𝑘 / 𝑛𝐴
4745, 46nffv 6673 . . . . . . . . . 10 𝑛(vol‘𝑘 / 𝑛𝐴)
4847nfeq1 2991 . . . . . . . . 9 𝑛(vol‘𝑘 / 𝑛𝐴) = +∞
49 csbeq1a 3895 . . . . . . . . . 10 (𝑛 = 𝑘𝐴 = 𝑘 / 𝑛𝐴)
5049fveqeq2d 6671 . . . . . . . . 9 (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘𝑘 / 𝑛𝐴) = +∞))
5144, 48, 50cbvrexw 3441 . . . . . . . 8 (∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ↔ ∃𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) = +∞)
5243, 51sylib 220 . . . . . . 7 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) = +∞)
5346nfel1 2992 . . . . . . . . . . . . 13 𝑛𝑘 / 𝑛𝐴 ∈ dom vol
5449eleq1d 2895 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ 𝑘 / 𝑛𝐴 ∈ dom vol))
5553, 54rspc 3609 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑘 / 𝑛𝐴 ∈ dom vol))
5655impcom 410 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑘 / 𝑛𝐴 ∈ dom vol)
57 iunmbl 24146 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
5857adantr 483 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
59 nfcv 2975 . . . . . . . . . . . . 13 𝑛
60 nfcv 2975 . . . . . . . . . . . . 13 𝑛𝑘
6159, 60, 46, 49ssiun2sf 30303 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴)
6261adantl 484 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴)
63 volss 24126 . . . . . . . . . . 11 ((𝑘 / 𝑛𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6456, 58, 62, 63syl3anc 1366 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6564adantlr 713 . . . . . . . . 9 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6665adantlr 713 . . . . . . . 8 ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑘 ∈ ℕ) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6766ralrimiva 3180 . . . . . . 7 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∀𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
68 r19.29r 3253 . . . . . . 7 ((∃𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) = +∞ ∧ ∀𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)) → ∃𝑘 ∈ ℕ ((vol‘𝑘 / 𝑛𝐴) = +∞ ∧ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)))
6952, 67, 68syl2anc 586 . . . . . 6 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ ((vol‘𝑘 / 𝑛𝐴) = +∞ ∧ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)))
70 breq1 5060 . . . . . . . 8 ((vol‘𝑘 / 𝑛𝐴) = +∞ → ((vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)))
7170biimpa 479 . . . . . . 7 (((vol‘𝑘 / 𝑛𝐴) = +∞ ∧ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
7271reximi 3241 . . . . . 6 (∃𝑘 ∈ ℕ ((vol‘𝑘 / 𝑛𝐴) = +∞ ∧ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)) → ∃𝑘 ∈ ℕ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
7369, 72syl 17 . . . . 5 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
74 1nn 11641 . . . . . 6 1 ∈ ℕ
75 ne0i 4298 . . . . . 6 (1 ∈ ℕ → ℕ ≠ ∅)
76 r19.9rzv 4443 . . . . . 6 (ℕ ≠ ∅ → (+∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ ∃𝑘 ∈ ℕ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)))
7774, 75, 76mp2b 10 . . . . 5 (+∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ ∃𝑘 ∈ ℕ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
7873, 77sylibr 236 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
79 iccssxr 12811 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
8012ffvelrni 6843 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
8179, 80sseldi 3963 . . . . . . 7 ( 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
8257, 81syl 17 . . . . . 6 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
8382ad2antrr 724 . . . . 5 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
84 xgepnf 12550 . . . . 5 ((vol‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ* → (+∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ (vol‘ 𝑛 ∈ ℕ 𝐴) = +∞))
8583, 84syl 17 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (+∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ (vol‘ 𝑛 ∈ ℕ 𝐴) = +∞))
8678, 85mpbid 234 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘ 𝑛 ∈ ℕ 𝐴) = +∞)
87 nfdisj1 5036 . . . . . 6 𝑛Disj 𝑛 ∈ ℕ 𝐴
887, 87nfan 1894 . . . . 5 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89 nfre1 3304 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘𝐴) = +∞
9088, 89nfan 1894 . . . 4 𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91 nnex 11636 . . . . 5 ℕ ∈ V
9291a1i 11 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ℕ ∈ V)
93143ad2antr3 1185 . . . . 5 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94933anassrs 1355 . . . 4 ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
9590, 92, 94, 43esumpinfval 31325 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
9686, 95eqtr4d 2857 . 2 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘ 𝑛 ∈ ℕ 𝐴) = Σ*𝑛 ∈ ℕ(vol‘𝐴))
97 exmid 891 . . . . 5 (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
98 rexnal 3236 . . . . . 6 (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ ↔ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
9998orbi2i 909 . . . . 5 ((∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
10097, 99mpbir 233 . . . 4 (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ)
101 r19.29 3252 . . . . . . 7 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102 xrge0nre 12833 . . . . . . . . 9 (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
10313, 102sylan 582 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104103reximi 3241 . . . . . . 7 (∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
105101, 104syl 17 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
106105ex 415 . . . . 5 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107106orim2d 963 . . . 4 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → ((∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)))
108100, 107mpi 20 . . 3 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
109108adantr 483 . 2 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
11042, 96, 109mpjaodan 955 1 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (vol‘ 𝑛 ∈ ℕ 𝐴) = Σ*𝑛 ∈ ℕ(vol‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wo 843  w3a 1082   = wceq 1531  wcel 2108  wne 3014  wral 3136  wrex 3137  Vcvv 3493  csb 3881  wss 3934  c0 4289   ciun 4910  Disj wdisj 5022   class class class wbr 5057  cmpt 5137  dom cdm 5548  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7148  supcsup 8896  cr 10528  0cc0 10529  1c1 10530   + caddc 10532  +∞cpnf 10664  *cxr 10666   < clt 10667  cle 10668  cn 11630  [,)cico 12732  [,]cicc 12733  seqcseq 13361  volcvol 24056  Σ*cesum 31279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cc 9849  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-of 7401  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-2o 8095  df-oadd 8098  df-er 8281  df-map 8400  df-pm 8401  df-ixp 8454  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-fi 8867  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-q 12341  df-rp 12382  df-xneg 12499  df-xadd 12500  df-xmul 12501  df-ioo 12734  df-ioc 12735  df-ico 12736  df-icc 12737  df-fz 12885  df-fzo 13026  df-fl 13154  df-mod 13230  df-seq 13362  df-exp 13422  df-fac 13626  df-bc 13655  df-hash 13683  df-shft 14418  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035  df-ef 15413  df-sin 15415  df-cos 15416  df-pi 15418  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-starv 16572  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-unif 16580  df-hom 16581  df-cco 16582  df-rest 16688  df-topn 16689  df-0g 16707  df-gsum 16708  df-topgen 16709  df-pt 16710  df-prds 16713  df-ordt 16766  df-xrs 16767  df-qtop 16772  df-imas 16773  df-xps 16775  df-mre 16849  df-mrc 16850  df-acs 16852  df-ps 17802  df-tsr 17803  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19232  df-ur 19244  df-ring 19291  df-cring 19292  df-subrg 19525  df-abv 19580  df-lmod 19628  df-scaf 19629  df-sra 19936  df-rgmod 19937  df-psmet 20529  df-xmet 20530  df-met 20531  df-bl 20532  df-mopn 20533  df-fbas 20534  df-fg 20535  df-cnfld 20538  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cld 21619  df-ntr 21620  df-cls 21621  df-nei 21698  df-lp 21736  df-perf 21737  df-cn 21827  df-cnp 21828  df-haus 21915  df-tx 22162  df-hmeo 22355  df-fil 22446  df-fm 22538  df-flim 22539  df-flf 22540  df-tmd 22672  df-tgp 22673  df-tsms 22727  df-trg 22760  df-xms 22922  df-ms 22923  df-tms 22924  df-nm 23184  df-ngp 23185  df-nrg 23187  df-nlm 23188  df-ii 23477  df-cncf 23478  df-ovol 24057  df-vol 24058  df-limc 24456  df-dv 24457  df-log 25132  df-esum 31280
This theorem is referenced by:  volmeas  31483
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