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Theorem voliune 33056
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 24951 and voliun 25000. (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 3110 . . . . 5 (∀𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ (vol‘𝐴) ∈ ℝ) ↔ (∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ))
2 eqid 2731 . . . . . 6 seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴))) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘𝐴)))
3 eqid 2731 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘𝐴)) = (𝑛 ∈ ℕ ↦ (vol‘𝐴))
42, 3voliun 25000 . . . . 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 3280 . . . . . . 7 𝑛𝑛 ∈ ℕ 𝐴 ∈ dom vol
8 nfra1 3280 . . . . . . 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 3244 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ dom vol)
12 volf 24975 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
1312ffvelcdmi 7070 . . . . . . . . . . 11 (𝐴 ∈ dom vol → (vol‘𝐴) ∈ (0[,]+∞))
1411, 13syl 17 . . . . . . . . . 10 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ 𝑛 ∈ ℕ) → (vol‘𝐴) ∈ (0[,]+∞))
153fvmpt2 6995 . . . . . . . . . 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 3253 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝑛 ∈ ℕ ↦ (vol‘𝐴))‘𝑛) = (vol‘𝐴))
209, 19esumeq2d 32864 . . . . 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 3247 . . . . . . . 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 11243 . . . . . . . . . . 11 0 ∈ ℝ*
25 pnfxr 11250 . . . . . . . . . . 11 +∞ ∈ ℝ*
26 elicc1 13350 . . . . . . . . . . 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 13082 . . . . . . . . 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 11198 . . . . . . . . 9 0 ∈ ℝ
33 elico2 13370 . . . . . . . . 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 7101 . . . . . 6 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∀𝑛 ∈ ℕ (vol‘𝐴) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘𝐴)):ℕ⟶(0[,)+∞))
37 nfmpt1 5249 . . . . . . 7 𝑛(𝑛 ∈ ℕ ↦ (vol‘𝐴))
3837esumfsupre 32898 . . . . . 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 2773 . . . 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 2774 . 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 2902 . . . . . . . . . . 11 𝑛vol
46 nfcsb1v 3914 . . . . . . . . . . 11 𝑛𝑘 / 𝑛𝐴
4745, 46nffv 6888 . . . . . . . . . 10 𝑛(vol‘𝑘 / 𝑛𝐴)
4847nfeq1 2917 . . . . . . . . 9 𝑛(vol‘𝑘 / 𝑛𝐴) = +∞
49 csbeq1a 3903 . . . . . . . . . 10 (𝑛 = 𝑘𝐴 = 𝑘 / 𝑛𝐴)
5049fveqeq2d 6886 . . . . . . . . 9 (𝑛 = 𝑘 → ((vol‘𝐴) = +∞ ↔ (vol‘𝑘 / 𝑛𝐴) = +∞))
5144, 48, 50cbvrexw 3303 . . . . . . . 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 2918 . . . . . . . . . . . . 13 𝑛𝑘 / 𝑛𝐴 ∈ dom vol
5449eleq1d 2817 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → (𝐴 ∈ dom vol ↔ 𝑘 / 𝑛𝐴 ∈ dom vol))
5553, 54rspc 3597 . . . . . . . . . . . 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 24999 . . . . . . . . . . . 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 2902 . . . . . . . . . . . . 13 𝑛
60 nfcv 2902 . . . . . . . . . . . . 13 𝑛𝑘
6159, 60, 46, 49ssiun2sf 31656 . . . . . . . . . . . 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 24979 . . . . . . . . . . 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 3145 . . . . . . 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 3115 . . . . . . 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 5144 . . . . . . . 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 3083 . . . . . 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 12205 . . . . . 6 1 ∈ ℕ
75 ne0i 4330 . . . . . 6 (1 ∈ ℕ → ℕ ≠ ∅)
76 r19.9rzv 4493 . . . . . 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 13389 . . . . . . . 8 (0[,]+∞) ⊆ ℝ*
8012ffvelcdmi 7070 . . . . . . . 8 ( 𝑛 ∈ ℕ 𝐴 ∈ dom vol → (vol‘ 𝑛 ∈ ℕ 𝐴) ∈ (0[,]+∞))
8179, 80sselid 3976 . . . . . . 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 13126 . . . . 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 5120 . . . . . 6 𝑛Disj 𝑛 ∈ ℕ 𝐴
887, 87nfan 1902 . . . . 5 𝑛(∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴)
89 nfre1 3281 . . . . 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 12200 . . . . 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 32900 . . 3 (((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ Disj 𝑛 ∈ ℕ 𝐴) ∧ ∃𝑛 ∈ ℕ (vol‘𝐴) = +∞) → Σ*𝑛 ∈ ℕ(vol‘𝐴) = +∞)
9686, 95eqtr4d 2774 . 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 3099 . . . . . 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 3113 . . . . . . 7 ((∀𝑛 ∈ ℕ 𝐴 ∈ dom vol ∧ ∃𝑛 ∈ ℕ ¬ (vol‘𝐴) ∈ ℝ) → ∃𝑛 ∈ ℕ (𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ))
102 xrge0nre 13412 . . . . . . . . 9 (((vol‘𝐴) ∈ (0[,]+∞) ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
10313, 102sylan 580 . . . . . . . 8 ((𝐴 ∈ dom vol ∧ ¬ (vol‘𝐴) ∈ ℝ) → (vol‘𝐴) = +∞)
104103reximi 3083 . . . . . . 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 2939  wral 3060  wrex 3069  Vcvv 3473  csb 3889  wss 3944  c0 4318   ciun 4990  Disj wdisj 5106   class class class wbr 5141  cmpt 5224  dom cdm 5669  ran crn 5670  wf 6528  cfv 6532  (class class class)co 7393  supcsup 9417  cr 11091  0cc0 11092  1c1 11093   + caddc 11095  +∞cpnf 11227  *cxr 11229   < clt 11230  cle 11231  cn 12194  [,)cico 13308  [,]cicc 13309  seqcseq 13948  volcvol 24909  Σ*cesum 32854
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-inf2 9618  ax-cc 10412  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170  ax-addf 11171  ax-mulf 11172
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-supp 8129  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-2o 8449  df-oadd 8452  df-er 8686  df-map 8805  df-pm 8806  df-ixp 8875  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-fsupp 9345  df-fi 9388  df-sup 9419  df-inf 9420  df-oi 9487  df-dju 9878  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-div 11854  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-7 12262  df-8 12263  df-9 12264  df-n0 12455  df-xnn0 12527  df-z 12541  df-dec 12660  df-uz 12805  df-q 12915  df-rp 12957  df-xneg 13074  df-xadd 13075  df-xmul 13076  df-ioo 13310  df-ioc 13311  df-ico 13312  df-icc 13313  df-fz 13467  df-fzo 13610  df-fl 13739  df-mod 13817  df-seq 13949  df-exp 14010  df-fac 14216  df-bc 14245  df-hash 14273  df-shft 14996  df-cj 15028  df-re 15029  df-im 15030  df-sqrt 15164  df-abs 15165  df-limsup 15397  df-clim 15414  df-rlim 15415  df-sum 15615  df-ef 15993  df-sin 15995  df-cos 15996  df-pi 15998  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-starv 17194  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-unif 17202  df-hom 17203  df-cco 17204  df-rest 17350  df-topn 17351  df-0g 17369  df-gsum 17370  df-topgen 17371  df-pt 17372  df-prds 17375  df-ordt 17429  df-xrs 17430  df-qtop 17435  df-imas 17436  df-xps 17438  df-mre 17512  df-mrc 17513  df-acs 17515  df-ps 18501  df-tsr 18502  df-plusf 18542  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-mhm 18647  df-submnd 18648  df-grp 18797  df-minusg 18798  df-sbg 18799  df-mulg 18923  df-subg 18975  df-cntz 19147  df-cmn 19614  df-abl 19615  df-mgp 19947  df-ur 19964  df-ring 20016  df-cring 20017  df-subrg 20310  df-abv 20374  df-lmod 20422  df-scaf 20423  df-sra 20734  df-rgmod 20735  df-psmet 20870  df-xmet 20871  df-met 20872  df-bl 20873  df-mopn 20874  df-fbas 20875  df-fg 20876  df-cnfld 20879  df-top 22325  df-topon 22342  df-topsp 22364  df-bases 22378  df-cld 22452  df-ntr 22453  df-cls 22454  df-nei 22531  df-lp 22569  df-perf 22570  df-cn 22660  df-cnp 22661  df-haus 22748  df-tx 22995  df-hmeo 23188  df-fil 23279  df-fm 23371  df-flim 23372  df-flf 23373  df-tmd 23505  df-tgp 23506  df-tsms 23560  df-trg 23593  df-xms 23755  df-ms 23756  df-tms 23757  df-nm 24020  df-ngp 24021  df-nrg 24023  df-nlm 24024  df-ii 24322  df-cncf 24323  df-ovol 24910  df-vol 24911  df-limc 25312  df-dv 25313  df-log 25994  df-esum 32855
This theorem is referenced by:  volmeas  33058
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