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Theorem voliune 32892
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 24892 and voliun 24941. (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 3111 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2 eqid 2733 . . . . . 6 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3 eqid 2733 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
42, 3voliun 24941 . . . . 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 583 . . . 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 651 . . 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 3266 . . . . . . 7 𝑛𝑛 ∈ ℕ 𝐴 ∈ dom vol
8 nfra1 3266 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
97, 8nfan 1903 . . . . . 6 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10 simpr 486 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11 rspa 3230 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12 volf 24916 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
1312ffvelcdmi 7038 . . . . . . . . . . 11 (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
1411, 13syl 17 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
153fvmpt2 6963 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1610, 14, 15syl2anc 585 . . . . . . . . 9 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1716adantlr 714 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1817ex 414 . . . . . . 7 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
199, 18ralrimi 3239 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
209, 19esumeq2d 32700 . . . . 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 486 . . . . . . . . 9 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
2221r19.21bi 3233 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ ℝ)
2314adantlr 714 . . . . . . . . 9 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
24 0xr 11210 . . . . . . . . . . 11 0 ∈ ℝ*
25 pnfxr 11217 . . . . . . . . . . 11 +∞ ∈ ℝ*
26 elicc1 13317 . . . . . . . . . . 11 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞)))
2724, 25, 26mp2an 691 . . . . . . . . . 10 ((vol‘𝐴) ∈ (0[,]+∞) ↔ ((vol‘𝐴) ∈ ℝ* ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) ≤ +∞))
2827simp2bi 1147 . . . . . . . . 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 13049 . . . . . . . . 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 11165 . . . . . . . . 9 0 ∈ ℝ
33 elico2 13337 . . . . . . . . 9 ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞)))
3432, 25, 33mp2an 691 . . . . . . . 8 ((vol‘𝐴) ∈ (0[,)+∞) ↔ ((vol‘𝐴) ∈ ℝ ∧ 0 ≤ (vol‘𝐴) ∧ (vol‘𝐴) < +∞))
3522, 29, 31, 34syl3anbrc 1344 . . . . . . 7 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
369, 35, 3fmptdf 7069 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37 nfmpt1 5217 . . . . . . 7 𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
3837esumfsupre 32734 . . . . . 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 2775 . . . 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 714 . . 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 2776 . 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 486 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44 nfv 1918 . . . . . . . . 9 𝑘(vol‘𝐴) = +∞
45 nfcv 2904 . . . . . . . . . . 11 𝑛vol
46 nfcsb1v 3884 . . . . . . . . . . 11 𝑛𝑘 / 𝑛𝐴
4745, 46nffv 6856 . . . . . . . . . 10 𝑛(vol‘𝑘 / 𝑛𝐴)
4847nfeq1 2919 . . . . . . . . 9 𝑛(vol‘𝑘 / 𝑛𝐴) = +∞
49 csbeq1a 3873 . . . . . . . . . 10 (𝑛 = 𝑘𝐴 = 𝑘 / 𝑛𝐴)
5049fveqeq2d 6854 . . . . . . . . 9 (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘𝑘 / 𝑛𝐴) = +∞))
5144, 48, 50cbvrexw 3289 . . . . . . . 8 (∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ↔ ∃𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) = +∞)
5243, 51sylib 217 . . . . . . 7 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑘 ∈ ℕ (vol‘𝑘 / 𝑛𝐴) = +∞)
5346nfel1 2920 . . . . . . . . . . . . 13 𝑛𝑘 / 𝑛𝐴 ∈ dom vol
5449eleq1d 2819 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ 𝑘 / 𝑛𝐴 ∈ dom vol))
5553, 54rspc 3571 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑘 / 𝑛𝐴 ∈ dom vol))
5655impcom 409 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑘 / 𝑛𝐴 ∈ dom vol)
57 iunmbl 24940 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
5857adantr 482 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
59 nfcv 2904 . . . . . . . . . . . . 13 𝑛
60 nfcv 2904 . . . . . . . . . . . . 13 𝑛𝑘
6159, 60, 46, 49ssiun2sf 31531 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴)
6261adantl 483 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴)
63 volss 24920 . . . . . . . . . . 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 1372 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6564adantlr 714 . . . . . . . . 9 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ 𝑘 ∈ ℕ) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6665adantlr 714 . . . . . . . 8 ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑘 ∈ ℕ) → (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
6766ralrimiva 3140 . . . . . . 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 3116 . . . . . . 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 585 . . . . . 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 5112 . . . . . . . 8 ((vol‘𝑘 / 𝑛𝐴) = +∞ → ((vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)))
7170biimpa 478 . . . . . . 7 (((vol‘𝑘 / 𝑛𝐴) = +∞ ∧ (vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
7271reximi 3084 . . . . . 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 12172 . . . . . 6 1 ∈ ℕ
75 ne0i 4298 . . . . . 6 (1 ∈ ℕ → ℕ ≠ ∅)
76 r19.9rzv 4461 . . . . . 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 233 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴))
79 iccssxr 13356 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
8012ffvelcdmi 7038 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
8179, 80sselid 3946 . . . . . . 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 725 . . . . 5 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ ℝ*)
84 xgepnf 13093 . . . . 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 231 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → (vol‘ 𝑛 ∈ ℕ 𝐴) = +∞)
87 nfdisj1 5088 . . . . . 6 𝑛Disj 𝑛 ∈ ℕ 𝐴
887, 87nfan 1903 . . . . 5 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89 nfre1 3267 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘𝐴) = +∞
9088, 89nfan 1903 . . . 4 𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91 nnex 12167 . . . . 5 ℕ ∈ V
9291a1i 11 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ℕ ∈ V)
93143ad2antr3 1191 . . . . 5 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94933anassrs 1361 . . . 4 ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
9590, 92, 94, 43esumpinfval 32736 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
9686, 95eqtr4d 2776 . 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 894 . . . . 5 (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
98 rexnal 3100 . . . . . 6 (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ ↔ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
9998orbi2i 912 . . . . 5 ((∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ¬ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
10097, 99mpbir 230 . . . 4 (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ)
101 r19.29 3114 . . . . . . 7 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102 xrge0nre 13379 . . . . . . . . 9 (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
10313, 102sylan 581 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104103reximi 3084 . . . . . . 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 414 . . . . 5 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107106orim2d 966 . . . 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 482 . 2 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
11042, 96, 109mpjaodan 958 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 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  Vcvv 3447  csb 3859  wss 3914  c0 4286   ciun 4958  Disj wdisj 5074   class class class wbr 5109  cmpt 5192  dom cdm 5637  ran crn 5638  wf 6496  cfv 6500  (class class class)co 7361  supcsup 9384  cr 11058  0cc0 11059  1c1 11060   + caddc 11062  +∞cpnf 11194  *cxr 11196   < clt 11197  cle 11198  cn 12161  [,)cico 13275  [,]cicc 13276  seqcseq 13915  volcvol 24850  Σ*cesum 32690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585  ax-cc 10379  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137  ax-addf 11138  ax-mulf 11139
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-iin 4961  df-disj 5075  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-of 7621  df-om 7807  df-1st 7925  df-2nd 7926  df-supp 8097  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-er 8654  df-map 8773  df-pm 8774  df-ixp 8842  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-fsupp 9312  df-fi 9355  df-sup 9386  df-inf 9387  df-oi 9454  df-dju 9845  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-xnn0 12494  df-z 12508  df-dec 12627  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-ioo 13277  df-ioc 13278  df-ico 13279  df-icc 13280  df-fz 13434  df-fzo 13577  df-fl 13706  df-mod 13784  df-seq 13916  df-exp 13977  df-fac 14183  df-bc 14212  df-hash 14240  df-shft 14961  df-cj 14993  df-re 14994  df-im 14995  df-sqrt 15129  df-abs 15130  df-limsup 15362  df-clim 15379  df-rlim 15380  df-sum 15580  df-ef 15958  df-sin 15960  df-cos 15961  df-pi 15963  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-starv 17156  df-sca 17157  df-vsca 17158  df-ip 17159  df-tset 17160  df-ple 17161  df-ds 17163  df-unif 17164  df-hom 17165  df-cco 17166  df-rest 17312  df-topn 17313  df-0g 17331  df-gsum 17332  df-topgen 17333  df-pt 17334  df-prds 17337  df-ordt 17391  df-xrs 17392  df-qtop 17397  df-imas 17398  df-xps 17400  df-mre 17474  df-mrc 17475  df-acs 17477  df-ps 18463  df-tsr 18464  df-plusf 18504  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-mhm 18609  df-submnd 18610  df-grp 18759  df-minusg 18760  df-sbg 18761  df-mulg 18881  df-subg 18933  df-cntz 19105  df-cmn 19572  df-abl 19573  df-mgp 19905  df-ur 19922  df-ring 19974  df-cring 19975  df-subrg 20262  df-abv 20319  df-lmod 20367  df-scaf 20368  df-sra 20678  df-rgmod 20679  df-psmet 20811  df-xmet 20812  df-met 20813  df-bl 20814  df-mopn 20815  df-fbas 20816  df-fg 20817  df-cnfld 20820  df-top 22266  df-topon 22283  df-topsp 22305  df-bases 22319  df-cld 22393  df-ntr 22394  df-cls 22395  df-nei 22472  df-lp 22510  df-perf 22511  df-cn 22601  df-cnp 22602  df-haus 22689  df-tx 22936  df-hmeo 23129  df-fil 23220  df-fm 23312  df-flim 23313  df-flf 23314  df-tmd 23446  df-tgp 23447  df-tsms 23501  df-trg 23534  df-xms 23696  df-ms 23697  df-tms 23698  df-nm 23961  df-ngp 23962  df-nrg 23964  df-nlm 23965  df-ii 24263  df-cncf 24264  df-ovol 24851  df-vol 24852  df-limc 25253  df-dv 25254  df-log 25935  df-esum 32691
This theorem is referenced by:  volmeas  32894
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