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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  voliunsge0lem Structured version   Visualization version   GIF version

Theorem voliunsge0lem 44787
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 1918 . . . . 5 𝑛𝜑
2 nfcv 2908 . . . . . . 7 𝑛vol
3 nfiu1 4993 . . . . . . 7 𝑛 𝑛 ∈ ℕ (𝐸𝑛)
42, 3nffv 6857 . . . . . 6 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛))
54nfeq1 2923 . . . . 5 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞
6 iccssxr 13354 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
7 volf 24909 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
87a1i 11 . . . . . . . . . . 11 (𝜑 → vol:dom vol⟶(0[,]+∞))
9 voliunsge0lem.e . . . . . . . . . . . . . 14 (𝜑𝐸:ℕ⟶dom vol)
109ffvelcdmda 7040 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
1110ralrimiva 3144 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
12 iunmbl 24933 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
1311, 12syl 17 . . . . . . . . . . 11 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
148, 13ffvelcdmd 7041 . . . . . . . . . 10 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ (0[,]+∞))
156, 14sselid 3947 . . . . . . . . 9 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
1615adantr 482 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17163adant3 1133 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
18 id 22 . . . . . . . . . 10 ((vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) = +∞)
1918eqcomd 2743 . . . . . . . . 9 ((vol‘(𝐸𝑛)) = +∞ → +∞ = (vol‘(𝐸𝑛)))
20193ad2ant3 1136 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ = (vol‘(𝐸𝑛)))
2113adantr 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
22 ssiun2 5012 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
2322adantl 483 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
24 volss 24913 . . . . . . . . . 10 (((𝐸𝑛) ∈ dom vol ∧ 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol ∧ (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2510, 21, 23, 24syl3anc 1372 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
26253adant3 1133 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2720, 26eqbrtrd 5132 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2817, 27xrgepnfd 43639 . . . . . 6 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
29283exp 1120 . . . . 5 (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)))
301, 5, 29rexlimd 3252 . . . 4 (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞))
3130imp 408 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
32 nfre1 3271 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞
331, 32nfan 1903 . . . 4 𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
34 nnex 12166 . . . . 5 ℕ ∈ V
3534a1i 11 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ℕ ∈ V)
367a1i 11 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3736, 10ffvelcdmd 7041 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
3837adantlr 714 . . . 4 (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
39 simpr 486 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4033, 35, 38, 39sge0pnfmpt 44760 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = +∞)
4131, 40eqtr4d 2780 . 2 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
42 ralnex 3076 . . . . . 6 (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4342biimpri 227 . . . . 5 (¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4443adantl 483 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4537adantr 482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
4618necon3bi 2971 . . . . . . . . . 10 (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ≠ +∞)
4746adantl 483 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≠ +∞)
48 ge0xrre 43843 . . . . . . . . 9 (((vol‘(𝐸𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸𝑛)) ≠ +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
4945, 47, 48syl2anc 585 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
5049ex 414 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ∈ ℝ))
51 renepnf 11210 . . . . . . . . 9 ((vol‘(𝐸𝑛)) ∈ ℝ → (vol‘(𝐸𝑛)) ≠ +∞)
5251neneqd 2949 . . . . . . . 8 ((vol‘(𝐸𝑛)) ∈ ℝ → ¬ (vol‘(𝐸𝑛)) = +∞)
5352a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((vol‘(𝐸𝑛)) ∈ ℝ → ¬ (vol‘(𝐸𝑛)) = +∞))
5450, 53impbid 211 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ ↔ (vol‘(𝐸𝑛)) ∈ ℝ))
5554ralbidva 3173 . . . . 5 (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5655adantr 482 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5744, 56mpbid 231 . . 3 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
58 nfra1 3270 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ
591, 58nfan 1903 . . . . . 6 𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
6010adantlr 714 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
61 rspa 3234 . . . . . . . . 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 513 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
6463ex 414 . . . . . 6 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ)))
6559, 64ralrimi 3243 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
66 voliunsge0lem.d . . . . . 6 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
6766adantr 482 . . . . 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 24934 . . . . 5 ((∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
7165, 67, 70syl2anc 585 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
72 1zzd 12541 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → 1 ∈ ℤ)
73 nnuz 12813 . . . . 5 ℕ = (ℤ‘1)
74 nfv 1918 . . . . . . . . 9 𝑛 𝑚 ∈ ℕ
7559, 74nfan 1903 . . . . . . . 8 𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
76 nfv 1918 . . . . . . . 8 𝑛(vol‘(𝐸𝑚)) ∈ (0[,)+∞)
7775, 76nfim 1900 . . . . . . 7 𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
78 eleq1w 2821 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
7978anbi2d 630 . . . . . . . 8 (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
80 2fveq3 6852 . . . . . . . . 9 (𝑛 = 𝑚 → (vol‘(𝐸𝑛)) = (vol‘(𝐸𝑚)))
8180eleq1d 2823 . . . . . . . 8 (𝑛 = 𝑚 → ((vol‘(𝐸𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸𝑚)) ∈ (0[,)+∞)))
8279, 81imbi12d 345 . . . . . . 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 11209 . . . . . . . . 9 0 ∈ ℝ*
8483a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*)
85 pnfxr 11216 . . . . . . . . 9 +∞ ∈ ℝ*
8685a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*)
8762rexrd 11212 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ*)
88 volge0 44276 . . . . . . . . . 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 13049 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) < +∞)
9284, 86, 87, 90, 91elicod 13321 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,)+∞))
9377, 82, 92chvarfv 2234 . . . . . 6 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
9480cbvmptv 5223 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸𝑚)))
9593, 94fmptd 7067 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))):ℕ⟶(0[,)+∞))
96 seqeq3 13918 . . . . . . 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 2765 . . . . 5 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))))
9972, 73, 95, 98sge0seq 44761 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = sup(ran 𝑆, ℝ*, < ))
10071, 99eqtr4d 2780 . . 3 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
10157, 100syldan 592 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  wss 3915   ciun 4959  Disj wdisj 5075   class class class wbr 5110  cmpt 5193  dom cdm 5638  ran crn 5639  wf 6497  cfv 6501  (class class class)co 7362  supcsup 9383  cr 11057  0cc0 11058  1c1 11059   + caddc 11061  +∞cpnf 11193  *cxr 11195   < clt 11196  cle 11197  cn 12160  [,)cico 13273  [,]cicc 13274  seqcseq 13913  volcvol 24843  Σ^csumge0 44677
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584  ax-cc 10378  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7622  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9385  df-inf 9386  df-oi 9453  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-xadd 13041  df-ioo 13275  df-ico 13277  df-icc 13278  df-fz 13432  df-fzo 13575  df-fl 13704  df-seq 13914  df-exp 13975  df-hash 14238  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-clim 15377  df-rlim 15378  df-sum 15578  df-xmet 20805  df-met 20806  df-ovol 24844  df-vol 24845  df-sumge0 44678
This theorem is referenced by:  voliunsge0  44788
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