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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  cr 10536  0cc0 10537  1c1 10538   + caddc 10540  +∞cpnf 10672  *cxr 10674   < clt 10675  cle 10676  cn 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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