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Theorem voliune 33513
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 25246 and voliun 25295. (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 2732 . . . . . 6 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3 eqid 2732 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
42, 3voliun 25295 . . . . 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 582 . . . 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 3281 . . . . . . 7 𝑛𝑛 ∈ ℕ 𝐴 ∈ dom vol
8 nfra1 3281 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ
97, 8nfan 1902 . . . . . 6 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
10 simpr 485 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
11 rspa 3245 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12 volf 25270 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
1312ffvelcdmi 7085 . . . . . . . . . . 11 (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
1411, 13syl 17 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
153fvmpt2 7009 . . . . . . . . . 10 ((𝑛 ∈ ℕ ∧ (vol‘𝐴) ∈ (0[,]+∞)) → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
1610, 14, 15syl2anc 584 . . . . . . . . 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 413 . . . . . . 7 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴)))
199, 18ralrimi 3254 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
209, 19esumeq2d 33321 . . . . 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 485 . . . . . . . . 9 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ)
2221r19.21bi 3248 . . . . . . . 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 11265 . . . . . . . . . . 11 0 ∈ ℝ*
25 pnfxr 11272 . . . . . . . . . . 11 +∞ ∈ ℝ*
26 elicc1 13372 . . . . . . . . . . 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 1146 . . . . . . . . 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 13104 . . . . . . . . 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 11220 . . . . . . . . 9 0 ∈ ℝ
33 elico2 13392 . . . . . . . . 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 1343 . . . . . . 7 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,)+∞))
369, 35, 3fmptdf 7118 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37 nfmpt1 5256 . . . . . . 7 𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
3837esumfsupre 33355 . . . . . 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 2774 . . . 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 2775 . 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 485 . . . . . . . 8 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
44 nfv 1917 . . . . . . . . 9 𝑘(vol‘𝐴) = +∞
45 nfcv 2903 . . . . . . . . . . 11 𝑛vol
46 nfcsb1v 3918 . . . . . . . . . . 11 𝑛𝑘 / 𝑛𝐴
4745, 46nffv 6901 . . . . . . . . . 10 𝑛(vol‘𝑘 / 𝑛𝐴)
4847nfeq1 2918 . . . . . . . . 9 𝑛(vol‘𝑘 / 𝑛𝐴) = +∞
49 csbeq1a 3907 . . . . . . . . . 10 (𝑛 = 𝑘𝐴 = 𝑘 / 𝑛𝐴)
5049fveqeq2d 6899 . . . . . . . . 9 (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘𝑘 / 𝑛𝐴) = +∞))
5144, 48, 50cbvrexw 3304 . . . . . . . 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 2919 . . . . . . . . . . . . 13 𝑛𝑘 / 𝑛𝐴 ∈ dom vol
5449eleq1d 2818 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ 𝑘 / 𝑛𝐴 ∈ dom vol))
5553, 54rspc 3600 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑘 / 𝑛𝐴 ∈ dom vol))
5655impcom 408 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑘 / 𝑛𝐴 ∈ dom vol)
57 iunmbl 25294 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
5857adantr 481 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑛 ∈ ℕ 𝐴 ∈ dom vol)
59 nfcv 2903 . . . . . . . . . . . . 13 𝑛
60 nfcv 2903 . . . . . . . . . . . . 13 𝑛𝑘
6159, 60, 46, 49ssiun2sf 32046 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴)
6261adantl 482 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑘 ∈ ℕ) → 𝑘 / 𝑛𝐴 𝑛 ∈ ℕ 𝐴)
63 volss 25274 . . . . . . . . . . 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 1371 . . . . . . . . . 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 3146 . . . . . . 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 584 . . . . . 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 5151 . . . . . . . 8 ((vol‘𝑘 / 𝑛𝐴) = +∞ → ((vol‘𝑘 / 𝑛𝐴) ≤ (vol‘ 𝑛 ∈ ℕ 𝐴) ↔ +∞ ≤ (vol‘ 𝑛 ∈ ℕ 𝐴)))
7170biimpa 477 . . . . . . 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 12227 . . . . . 6 1 ∈ ℕ
75 ne0i 4334 . . . . . 6 (1 ∈ ℕ → ℕ ≠ ∅)
76 r19.9rzv 4499 . . . . . 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 13411 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
8012ffvelcdmi 7085 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
8179, 80sselid 3980 . . . . . . 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 13148 . . . . 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 5127 . . . . . 6 𝑛Disj 𝑛 ∈ ℕ 𝐴
887, 87nfan 1902 . . . . 5 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89 nfre1 3282 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘𝐴) = +∞
9088, 89nfan 1902 . . . 4 𝑛((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞)
91 nnex 12222 . . . . 5 ℕ ∈ V
9291a1i 11 . . . 4 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → ℕ ∈ V)
93143ad2antr3 1190 . . . . 5 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ (Disj 𝑛 ∈ ℕ 𝐴 ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞ ∧ 𝑛 ∈ ℕ)) → (vol‘𝐴) ∈ (0[,]+∞))
94933anassrs 1360 . . . 4 ((((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
9590, 92, 94, 43esumpinfval 33357 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
9686, 95eqtr4d 2775 . 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 893 . . . . 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 911 . . . . 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 13434 . . . . . . . . 9 (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
10313, 102sylan 580 . . . . . . . 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 413 . . . . 5 (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol → (∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ → ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
107106orim2d 965 . . . 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 481 . 2 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) → (∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ ∨ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞))
11042, 96, 109mpjaodan 957 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 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  Vcvv 3474  csb 3893  wss 3948  c0 4322   ciun 4997  Disj wdisj 5113   class class class wbr 5148  cmpt 5231  dom cdm 5676  ran crn 5677  wf 6539  cfv 6543  (class class class)co 7411  supcsup 9437  cr 11111  0cc0 11112  1c1 11113   + caddc 11115  +∞cpnf 11249  *cxr 11251   < clt 11252  cle 11253  cn 12216  [,)cico 13330  [,]cicc 13331  seqcseq 13970  volcvol 25204  Σ*cesum 33311
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-oadd 8472  df-er 8705  df-map 8824  df-pm 8825  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-dju 9898  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ioc 13333  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-mod 13839  df-seq 13971  df-exp 14032  df-fac 14238  df-bc 14267  df-hash 14295  df-shft 15018  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-limsup 15419  df-clim 15436  df-rlim 15437  df-sum 15637  df-ef 16015  df-sin 16017  df-cos 16018  df-pi 16020  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-ordt 17451  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-ps 18523  df-tsr 18524  df-plusf 18564  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-cntz 19222  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-subrng 20434  df-subrg 20459  df-abv 20568  df-lmod 20616  df-scaf 20617  df-sra 20930  df-rgmod 20931  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-fbas 21141  df-fg 21142  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-lp 22860  df-perf 22861  df-cn 22951  df-cnp 22952  df-haus 23039  df-tx 23286  df-hmeo 23479  df-fil 23570  df-fm 23662  df-flim 23663  df-flf 23664  df-tmd 23796  df-tgp 23797  df-tsms 23851  df-trg 23884  df-xms 24046  df-ms 24047  df-tms 24048  df-nm 24311  df-ngp 24312  df-nrg 24314  df-nlm 24315  df-ii 24617  df-cncf 24618  df-ovol 25205  df-vol 25206  df-limc 25607  df-dv 25608  df-log 26289  df-esum 33312
This theorem is referenced by:  volmeas  33515
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