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Theorem voliunsge0lem 45783
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 1910 . . . . 5 𝑛𝜑
2 nfcv 2898 . . . . . . 7 𝑛vol
3 nfiu1 5025 . . . . . . 7 𝑛 𝑛 ∈ ℕ (𝐸𝑛)
42, 3nffv 6901 . . . . . 6 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛))
54nfeq1 2913 . . . . 5 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞
6 iccssxr 13431 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
7 volf 25445 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
87a1i 11 . . . . . . . . . . 11 (𝜑 → vol:dom vol⟶(0[,]+∞))
9 voliunsge0lem.e . . . . . . . . . . . . . 14 (𝜑𝐸:ℕ⟶dom vol)
109ffvelcdmda 7088 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
1110ralrimiva 3141 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
12 iunmbl 25469 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
1311, 12syl 17 . . . . . . . . . . 11 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
148, 13ffvelcdmd 7089 . . . . . . . . . 10 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ (0[,]+∞))
156, 14sselid 3976 . . . . . . . . 9 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
1615adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17163adant3 1130 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
18 id 22 . . . . . . . . . 10 ((vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) = +∞)
1918eqcomd 2733 . . . . . . . . 9 ((vol‘(𝐸𝑛)) = +∞ → +∞ = (vol‘(𝐸𝑛)))
20193ad2ant3 1133 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ = (vol‘(𝐸𝑛)))
2113adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
22 ssiun2 5044 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
2322adantl 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
24 volss 25449 . . . . . . . . . 10 (((𝐸𝑛) ∈ dom vol ∧ 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol ∧ (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2510, 21, 23, 24syl3anc 1369 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
26253adant3 1130 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2720, 26eqbrtrd 5164 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2817, 27xrgepnfd 44636 . . . . . 6 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
29283exp 1117 . . . . 5 (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)))
301, 5, 29rexlimd 3258 . . . 4 (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞))
3130imp 406 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
32 nfre1 3277 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞
331, 32nfan 1895 . . . 4 𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
34 nnex 12240 . . . . 5 ℕ ∈ V
3534a1i 11 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ℕ ∈ V)
367a1i 11 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3736, 10ffvelcdmd 7089 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
3837adantlr 714 . . . 4 (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
39 simpr 484 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4033, 35, 38, 39sge0pnfmpt 45756 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = +∞)
4131, 40eqtr4d 2770 . 2 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
42 ralnex 3067 . . . . . 6 (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4342biimpri 227 . . . . 5 (¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4443adantl 481 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4537adantr 480 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
4618necon3bi 2962 . . . . . . . . . 10 (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ≠ +∞)
4746adantl 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≠ +∞)
48 ge0xrre 44839 . . . . . . . . 9 (((vol‘(𝐸𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸𝑛)) ≠ +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
4945, 47, 48syl2anc 583 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
5049ex 412 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ∈ ℝ))
51 renepnf 11284 . . . . . . . . 9 ((vol‘(𝐸𝑛)) ∈ ℝ → (vol‘(𝐸𝑛)) ≠ +∞)
5251neneqd 2940 . . . . . . . 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 3170 . . . . 5 (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5655adantr 480 . . . 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 3276 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ
591, 58nfan 1895 . . . . . 6 𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
6010adantlr 714 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
61 rspa 3240 . . . . . . . . 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 511 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
6463ex 412 . . . . . 6 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ)))
6559, 64ralrimi 3249 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
66 voliunsge0lem.d . . . . . 6 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
6766adantr 480 . . . . 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 25470 . . . . 5 ((∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
7165, 67, 70syl2anc 583 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
72 1zzd 12615 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → 1 ∈ ℤ)
73 nnuz 12887 . . . . 5 ℕ = (ℤ‘1)
74 nfv 1910 . . . . . . . . 9 𝑛 𝑚 ∈ ℕ
7559, 74nfan 1895 . . . . . . . 8 𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
76 nfv 1910 . . . . . . . 8 𝑛(vol‘(𝐸𝑚)) ∈ (0[,)+∞)
7775, 76nfim 1892 . . . . . . 7 𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
78 eleq1w 2811 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
7978anbi2d 628 . . . . . . . 8 (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
80 2fveq3 6896 . . . . . . . . 9 (𝑛 = 𝑚 → (vol‘(𝐸𝑛)) = (vol‘(𝐸𝑚)))
8180eleq1d 2813 . . . . . . . 8 (𝑛 = 𝑚 → ((vol‘(𝐸𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸𝑚)) ∈ (0[,)+∞)))
8279, 81imbi12d 344 . . . . . . 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 11283 . . . . . . . . 9 0 ∈ ℝ*
8483a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*)
85 pnfxr 11290 . . . . . . . . 9 +∞ ∈ ℝ*
8685a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*)
8762rexrd 11286 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ*)
88 volge0 45272 . . . . . . . . . 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 13125 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) < +∞)
9284, 86, 87, 90, 91elicod 13398 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,)+∞))
9377, 82, 92chvarfv 2226 . . . . . 6 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
9480cbvmptv 5255 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸𝑚)))
9593, 94fmptd 7118 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))):ℕ⟶(0[,)+∞))
96 seqeq3 13995 . . . . . . 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 2755 . . . . 5 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))))
9972, 73, 95, 98sge0seq 45757 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = sup(ran 𝑆, ℝ*, < ))
10071, 99eqtr4d 2770 . . 3 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
10157, 100syldan 590 . 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 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  Vcvv 3469  wss 3944   ciun 4991  Disj wdisj 5107   class class class wbr 5142  cmpt 5225  dom cdm 5672  ran crn 5673  wf 6538  cfv 6542  (class class class)co 7414  supcsup 9455  cr 11129  0cc0 11130  1c1 11131   + caddc 11133  +∞cpnf 11267  *cxr 11269   < clt 11270  cle 11271  cn 12234  [,)cico 13350  [,]cicc 13351  seqcseq 13990  volcvol 25379  Σ^csumge0 45673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656  ax-cc 10450  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207  ax-pre-sup 11208
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-disj 5108  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-of 7679  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-oi 9525  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-div 11894  df-nn 12235  df-2 12297  df-3 12298  df-n0 12495  df-z 12581  df-uz 12845  df-q 12955  df-rp 12999  df-xadd 13117  df-ioo 13352  df-ico 13354  df-icc 13355  df-fz 13509  df-fzo 13652  df-fl 13781  df-seq 13991  df-exp 14051  df-hash 14314  df-cj 15070  df-re 15071  df-im 15072  df-sqrt 15206  df-abs 15207  df-clim 15456  df-rlim 15457  df-sum 15657  df-xmet 21259  df-met 21260  df-ovol 25380  df-vol 25381  df-sumge0 45674
This theorem is referenced by:  voliunsge0  45784
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