Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  voliunsge0lem Structured version   Visualization version   GIF version

Theorem voliunsge0lem 43064
 Description: The Lebesgue measure function is countably additive. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
voliunsge0lem.s 𝑆 = seq1( + , 𝐺)
voliunsge0lem.g 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))
voliunsge0lem.e (𝜑𝐸:ℕ⟶dom vol)
voliunsge0lem.d (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
Assertion
Ref Expression
voliunsge0lem (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
Distinct variable groups:   𝑛,𝐸   𝜑,𝑛
Allowed substitution hints:   𝑆(𝑛)   𝐺(𝑛)

Proof of Theorem voliunsge0lem
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 nfv 1916 . . . . 5 𝑛𝜑
2 nfcv 2982 . . . . . . 7 𝑛vol
3 nfiu1 4939 . . . . . . 7 𝑛 𝑛 ∈ ℕ (𝐸𝑛)
42, 3nffv 6673 . . . . . 6 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛))
54nfeq1 2997 . . . . 5 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞
6 iccssxr 12819 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
7 volf 24142 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
87a1i 11 . . . . . . . . . . 11 (𝜑 → vol:dom vol⟶(0[,]+∞))
9 voliunsge0lem.e . . . . . . . . . . . . . 14 (𝜑𝐸:ℕ⟶dom vol)
109ffvelrnda 6844 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
1110ralrimiva 3177 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
12 iunmbl 24166 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
1311, 12syl 17 . . . . . . . . . . 11 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
148, 13ffvelrnd 6845 . . . . . . . . . 10 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ (0[,]+∞))
156, 14sseldi 3951 . . . . . . . . 9 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
1615adantr 484 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17163adant3 1129 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
18 id 22 . . . . . . . . . 10 ((vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) = +∞)
1918eqcomd 2830 . . . . . . . . 9 ((vol‘(𝐸𝑛)) = +∞ → +∞ = (vol‘(𝐸𝑛)))
20193ad2ant3 1132 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ = (vol‘(𝐸𝑛)))
2113adantr 484 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
22 ssiun2 4957 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
2322adantl 485 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
24 volss 24146 . . . . . . . . . 10 (((𝐸𝑛) ∈ dom vol ∧ 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol ∧ (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2510, 21, 23, 24syl3anc 1368 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
26253adant3 1129 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2720, 26eqbrtrd 5075 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2817, 27xrgepnfd 41916 . . . . . 6 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
29283exp 1116 . . . . 5 (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)))
301, 5, 29rexlimd 3309 . . . 4 (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞))
3130imp 410 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
32 nfre1 3298 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞
331, 32nfan 1901 . . . 4 𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
34 nnex 11642 . . . . 5 ℕ ∈ V
3534a1i 11 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ℕ ∈ V)
367a1i 11 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3736, 10ffvelrnd 6845 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
3837adantlr 714 . . . 4 (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
39 simpr 488 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4033, 35, 38, 39sge0pnfmpt 43037 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = +∞)
4131, 40eqtr4d 2862 . 2 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
42 ralnex 3230 . . . . . 6 (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4342biimpri 231 . . . . 5 (¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4443adantl 485 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4537adantr 484 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
4618necon3bi 3040 . . . . . . . . . 10 (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ≠ +∞)
4746adantl 485 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≠ +∞)
48 ge0xrre 42121 . . . . . . . . 9 (((vol‘(𝐸𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸𝑛)) ≠ +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
4945, 47, 48syl2anc 587 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
5049ex 416 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ∈ ℝ))
51 renepnf 10689 . . . . . . . . 9 ((vol‘(𝐸𝑛)) ∈ ℝ → (vol‘(𝐸𝑛)) ≠ +∞)
5251neneqd 3019 . . . . . . . 8 ((vol‘(𝐸𝑛)) ∈ ℝ → ¬ (vol‘(𝐸𝑛)) = +∞)
5352a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((vol‘(𝐸𝑛)) ∈ ℝ → ¬ (vol‘(𝐸𝑛)) = +∞))
5450, 53impbid 215 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ ↔ (vol‘(𝐸𝑛)) ∈ ℝ))
5554ralbidva 3191 . . . . 5 (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5655adantr 484 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5744, 56mpbid 235 . . 3 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
58 nfra1 3213 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ
591, 58nfan 1901 . . . . . 6 𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
6010adantlr 714 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
61 rspa 3201 . . . . . . . . 9 ((∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ)
6261adantll 713 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ)
6360, 62jca 515 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
6463ex 416 . . . . . 6 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ)))
6559, 64ralrimi 3210 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
66 voliunsge0lem.d . . . . . 6 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
6766adantr 484 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → Disj 𝑛 ∈ ℕ (𝐸𝑛))
68 voliunsge0lem.s . . . . . 6 𝑆 = seq1( + , 𝐺)
69 voliunsge0lem.g . . . . . 6 𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))
7068, 69voliun 24167 . . . . 5 ((∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
7165, 67, 70syl2anc 587 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
72 1zzd 12012 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → 1 ∈ ℤ)
73 nnuz 12280 . . . . 5 ℕ = (ℤ‘1)
74 nfv 1916 . . . . . . . . 9 𝑛 𝑚 ∈ ℕ
7559, 74nfan 1901 . . . . . . . 8 𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
76 nfv 1916 . . . . . . . 8 𝑛(vol‘(𝐸𝑚)) ∈ (0[,)+∞)
7775, 76nfim 1898 . . . . . . 7 𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
78 eleq1w 2898 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
7978anbi2d 631 . . . . . . . 8 (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
80 2fveq3 6668 . . . . . . . . 9 (𝑛 = 𝑚 → (vol‘(𝐸𝑛)) = (vol‘(𝐸𝑚)))
8180eleq1d 2900 . . . . . . . 8 (𝑛 = 𝑚 → ((vol‘(𝐸𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸𝑚)) ∈ (0[,)+∞)))
8279, 81imbi12d 348 . . . . . . 7 (𝑛 = 𝑚 → ((((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,)+∞)) ↔ (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))))
83 0xr 10688 . . . . . . . . 9 0 ∈ ℝ*
8483a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*)
85 pnfxr 10695 . . . . . . . . 9 +∞ ∈ ℝ*
8685a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*)
8762rexrd 10691 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ*)
88 volge0 42556 . . . . . . . . . 10 ((𝐸𝑛) ∈ dom vol → 0 ≤ (vol‘(𝐸𝑛)))
8910, 88syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸𝑛)))
9089adantlr 714 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸𝑛)))
9162ltpnfd 12515 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) < +∞)
9284, 86, 87, 90, 91elicod 12786 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,)+∞))
9377, 82, 92chvarfv 2244 . . . . . 6 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
9480cbvmptv 5156 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸𝑚)))
9593, 94fmptd 6871 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))):ℕ⟶(0[,)+∞))
96 seqeq3 13380 . . . . . . 7 (𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
9769, 96ax-mp 5 . . . . . 6 seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))))
9868, 97eqtri 2847 . . . . 5 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))))
9972, 73, 95, 98sge0seq 43038 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = sup(ran 𝑆, ℝ*, < ))
10071, 99eqtr4d 2862 . . 3 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
10157, 100syldan 594 . 2 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
10241, 101pm2.61dan 812 1 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115   ≠ wne 3014  ∀wral 3133  ∃wrex 3134  Vcvv 3480   ⊆ wss 3919  ∪ ciun 4905  Disj wdisj 5018   class class class wbr 5053   ↦ cmpt 5133  dom cdm 5543  ran crn 5544  ⟶wf 6341  ‘cfv 6345  (class class class)co 7151  supcsup 8903  ℝcr 10536  0cc0 10537  1c1 10538   + caddc 10540  +∞cpnf 10672  ℝ*cxr 10674   < clt 10675   ≤ cle 10676  ℕcn 11636  [,)cico 12739  [,]cicc 12740  seqcseq 13375  volcvol 24076  Σ^csumge0 42954 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-inf2 9103  ax-cc 9857  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5019  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-se 5503  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-isom 6354  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7405  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-2o 8101  df-oadd 8104  df-er 8287  df-map 8406  df-pm 8407  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-sup 8905  df-inf 8906  df-oi 8973  df-dju 9329  df-card 9367  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11637  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-q 12348  df-rp 12389  df-xadd 12507  df-ioo 12741  df-ico 12743  df-icc 12744  df-fz 12897  df-fzo 13040  df-fl 13168  df-seq 13376  df-exp 13437  df-hash 13698  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-clim 14847  df-rlim 14848  df-sum 15045  df-xmet 20093  df-met 20094  df-ovol 24077  df-vol 24078  df-sumge0 42955 This theorem is referenced by:  voliunsge0  43065
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