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Theorem voliunsge0lem 39993
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 1840 . . . . 5 𝑛𝜑
2 nfcv 2761 . . . . . . 7 𝑛vol
3 nfiu1 4516 . . . . . . 7 𝑛 𝑛 ∈ ℕ (𝐸𝑛)
42, 3nffv 6155 . . . . . 6 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛))
54nfeq1 2774 . . . . 5 𝑛(vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞
6 iccssxr 12198 . . . . . . . . . 10 (0[,]+∞) ⊆ ℝ*
7 volf 23204 . . . . . . . . . . . 12 vol:dom vol⟶(0[,]+∞)
87a1i 11 . . . . . . . . . . 11 (𝜑 → vol:dom vol⟶(0[,]+∞))
9 voliunsge0lem.e . . . . . . . . . . . . . 14 (𝜑𝐸:ℕ⟶dom vol)
109ffvelrnda 6315 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
1110ralrimiva 2960 . . . . . . . . . . . 12 (𝜑 → ∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
12 iunmbl 23228 . . . . . . . . . . . 12 (∀𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
1311, 12syl 17 . . . . . . . . . . 11 (𝜑 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
148, 13ffvelrnd 6316 . . . . . . . . . 10 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ (0[,]+∞))
156, 14sseldi 3581 . . . . . . . . 9 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
1615adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
17163adant3 1079 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) ∈ ℝ*)
18 id 22 . . . . . . . . . 10 ((vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) = +∞)
1918eqcomd 2627 . . . . . . . . 9 ((vol‘(𝐸𝑛)) = +∞ → +∞ = (vol‘(𝐸𝑛)))
20193ad2ant3 1082 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ = (vol‘(𝐸𝑛)))
2113adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol)
22 ssiun2 4529 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
2322adantl 482 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛))
24 volss 23208 . . . . . . . . . 10 (((𝐸𝑛) ∈ dom vol ∧ 𝑛 ∈ ℕ (𝐸𝑛) ∈ dom vol ∧ (𝐸𝑛) ⊆ 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2510, 21, 23, 24syl3anc 1323 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
26253adant3 1079 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2720, 26eqbrtrd 4635 . . . . . . 7 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → +∞ ≤ (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)))
2817, 27xrgepnfd 39008 . . . . . 6 ((𝜑𝑛 ∈ ℕ ∧ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
29283exp 1261 . . . . 5 (𝜑 → (𝑛 ∈ ℕ → ((vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)))
301, 5, 29rexlimd 3019 . . . 4 (𝜑 → (∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞))
3130imp 445 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = +∞)
32 nfre1 2999 . . . . 5 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞
331, 32nfan 1825 . . . 4 𝑛(𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
34 nnex 10970 . . . . 5 ℕ ∈ V
3534a1i 11 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ℕ ∈ V)
367a1i 11 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → vol:dom vol⟶(0[,]+∞))
3736, 10ffvelrnd 6316 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
3837adantlr 750 . . . 4 (((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
39 simpr 477 . . . 4 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4033, 35, 38, 39sge0pnfmpt 39966 . . 3 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = +∞)
4131, 40eqtr4d 2658 . 2 ((𝜑 ∧ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
42 ralnex 2986 . . . . . 6 (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞)
4342biimpri 218 . . . . 5 (¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞ → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4443adantl 482 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞)
4537adantr 481 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ (0[,]+∞))
4618necon3bi 2816 . . . . . . . . . 10 (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ≠ +∞)
4746adantl 482 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ≠ +∞)
48 ge0xrre 39166 . . . . . . . . 9 (((vol‘(𝐸𝑛)) ∈ (0[,]+∞) ∧ (vol‘(𝐸𝑛)) ≠ +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
4945, 47, 48syl2anc 692 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ ¬ (vol‘(𝐸𝑛)) = +∞) → (vol‘(𝐸𝑛)) ∈ ℝ)
5049ex 450 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ → (vol‘(𝐸𝑛)) ∈ ℝ))
51 renepnf 10031 . . . . . . . . 9 ((vol‘(𝐸𝑛)) ∈ ℝ → (vol‘(𝐸𝑛)) ≠ +∞)
5251neneqd 2795 . . . . . . . 8 ((vol‘(𝐸𝑛)) ∈ ℝ → ¬ (vol‘(𝐸𝑛)) = +∞)
5352a1i 11 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((vol‘(𝐸𝑛)) ∈ ℝ → ¬ (vol‘(𝐸𝑛)) = +∞))
5450, 53impbid 202 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (¬ (vol‘(𝐸𝑛)) = +∞ ↔ (vol‘(𝐸𝑛)) ∈ ℝ))
5554ralbidva 2979 . . . . 5 (𝜑 → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5655adantr 481 . . . 4 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (∀𝑛 ∈ ℕ ¬ (vol‘(𝐸𝑛)) = +∞ ↔ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ))
5744, 56mpbid 222 . . 3 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
58 nfra1 2936 . . . . . . 7 𝑛𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ
591, 58nfan 1825 . . . . . 6 𝑛(𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ)
6010adantlr 750 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐸𝑛) ∈ dom vol)
61 rspa 2925 . . . . . . . . 9 ((∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ)
6261adantll 749 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ)
6360, 62jca 554 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
6463ex 450 . . . . . 6 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ → ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ)))
6559, 64ralrimi 2951 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → ∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ))
66 voliunsge0lem.d . . . . . 6 (𝜑Disj 𝑛 ∈ ℕ (𝐸𝑛))
6766adantr 481 . . . . 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 23229 . . . . 5 ((∀𝑛 ∈ ℕ ((𝐸𝑛) ∈ dom vol ∧ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ Disj 𝑛 ∈ ℕ (𝐸𝑛)) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
7165, 67, 70syl2anc 692 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = sup(ran 𝑆, ℝ*, < ))
72 1zzd 11352 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → 1 ∈ ℤ)
73 nnuz 11667 . . . . 5 ℕ = (ℤ‘1)
74 nfv 1840 . . . . . . . . 9 𝑛 𝑚 ∈ ℕ
7559, 74nfan 1825 . . . . . . . 8 𝑛((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)
76 nfv 1840 . . . . . . . 8 𝑛(vol‘(𝐸𝑚)) ∈ (0[,)+∞)
7775, 76nfim 1822 . . . . . . 7 𝑛(((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
78 eleq1 2686 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑛 ∈ ℕ ↔ 𝑚 ∈ ℕ))
7978anbi2d 739 . . . . . . . 8 (𝑛 = 𝑚 → (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) ↔ ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ)))
80 fveq2 6148 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐸𝑛) = (𝐸𝑚))
8180fveq2d 6152 . . . . . . . . 9 (𝑛 = 𝑚 → (vol‘(𝐸𝑛)) = (vol‘(𝐸𝑚)))
8281eleq1d 2683 . . . . . . . 8 (𝑛 = 𝑚 → ((vol‘(𝐸𝑛)) ∈ (0[,)+∞) ↔ (vol‘(𝐸𝑚)) ∈ (0[,)+∞)))
8379, 82imbi12d 334 . . . . . . 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[,)+∞))))
84 0xr 10030 . . . . . . . . 9 0 ∈ ℝ*
8584a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*)
86 pnfxr 10036 . . . . . . . . 9 +∞ ∈ ℝ*
8786a1i 11 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*)
8862rexrd 10033 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ ℝ*)
89 volge0 39481 . . . . . . . . . 10 ((𝐸𝑛) ∈ dom vol → 0 ≤ (vol‘(𝐸𝑛)))
9010, 89syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸𝑛)))
9190adantlr 750 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (vol‘(𝐸𝑛)))
9262ltpnfd 11899 . . . . . . . 8 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) < +∞)
9385, 87, 88, 91, 92elicod 12166 . . . . . . 7 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (vol‘(𝐸𝑛)) ∈ (0[,)+∞))
9477, 83, 93chvar 2261 . . . . . 6 (((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (vol‘(𝐸𝑚)) ∈ (0[,)+∞))
9581cbvmptv 4710 . . . . . 6 (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))) = (𝑚 ∈ ℕ ↦ (vol‘(𝐸𝑚)))
9694, 95fmptd 6340 . . . . 5 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))):ℕ⟶(0[,)+∞))
97 seqeq3 12746 . . . . . . 7 (𝐺 = (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))) → seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
9869, 97ax-mp 5 . . . . . 6 seq1( + , 𝐺) = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))))
9968, 98eqtri 2643 . . . . 5 𝑆 = seq1( + , (𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛))))
10072, 73, 96, 99sge0seq 39967 . . . 4 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))) = sup(ran 𝑆, ℝ*, < ))
10171, 100eqtr4d 2658 . . 3 ((𝜑 ∧ ∀𝑛 ∈ ℕ (vol‘(𝐸𝑛)) ∈ ℝ) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
10257, 101syldan 487 . 2 ((𝜑 ∧ ¬ ∃𝑛 ∈ ℕ (vol‘(𝐸𝑛)) = +∞) → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
10341, 102pm2.61dan 831 1 (𝜑 → (vol‘ 𝑛 ∈ ℕ (𝐸𝑛)) = (Σ^‘(𝑛 ∈ ℕ ↦ (vol‘(𝐸𝑛)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  wss 3555   ciun 4485  Disj wdisj 4583   class class class wbr 4613  cmpt 4673  dom cdm 5074  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  supcsup 8290  cr 9879  0cc0 9880  1c1 9881   + caddc 9883  +∞cpnf 10015  *cxr 10017   < clt 10018  cle 10019  cn 10964  [,)cico 12119  [,]cicc 12120  seqcseq 12741  volcvol 23139  Σ^csumge0 39883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cc 9201  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xadd 11891  df-ioo 12121  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-rlim 14154  df-sum 14351  df-xmet 19658  df-met 19659  df-ovol 23140  df-vol 23141  df-sumge0 39884
This theorem is referenced by:  voliunsge0  39994
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