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Theorem ovn0lem 43570
 Description: For any finite dimension, the Lebesgue outer measure of the empty set is zero. This is step (a)(ii) of the proof of Proposition 115D (a) of [Fremlin1] p. 30. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ovn0lem.x (𝜑𝑋 ∈ Fin)
ovn0lem.n0 (𝜑𝑋 ≠ ∅)
ovn0lem.m 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))}
ovn0lem.infm (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
ovn0lem.i 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨1, 0⟩))
Assertion
Ref Expression
ovn0lem (𝜑 → inf(𝑀, ℝ*, < ) = 0)
Distinct variable groups:   𝑖,𝐼,𝑗,𝑘   𝐼,𝑙,𝑗,𝑘   𝑖,𝑋,𝑗,𝑘,𝑧   𝑋,𝑙   𝜑,𝑗,𝑘,𝑙
Allowed substitution hints:   𝜑(𝑧,𝑖)   𝐼(𝑧)   𝑀(𝑧,𝑖,𝑗,𝑘,𝑙)

Proof of Theorem ovn0lem
StepHypRef Expression
1 iccssxr 12862 . . 3 (0[,]+∞) ⊆ ℝ*
2 ovn0lem.infm . . 3 (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
31, 2sseldi 3890 . 2 (𝜑 → inf(𝑀, ℝ*, < ) ∈ ℝ*)
4 0xr 10726 . . 3 0 ∈ ℝ*
54a1i 11 . 2 (𝜑 → 0 ∈ ℝ*)
6 ovn0lem.m . . . . 5 𝑀 = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))}
7 ssrab2 3984 . . . . 5 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))} ⊆ ℝ*
86, 7eqsstri 3926 . . . 4 𝑀 ⊆ ℝ*
98a1i 11 . . 3 (𝜑𝑀 ⊆ ℝ*)
10 1re 10679 . . . . . . . . . . . . . 14 1 ∈ ℝ
11 0re 10681 . . . . . . . . . . . . . 14 0 ∈ ℝ
1210, 11pm3.2i 474 . . . . . . . . . . . . 13 (1 ∈ ℝ ∧ 0 ∈ ℝ)
13 opelxp 5560 . . . . . . . . . . . . 13 (⟨1, 0⟩ ∈ (ℝ × ℝ) ↔ (1 ∈ ℝ ∧ 0 ∈ ℝ))
1412, 13mpbir 234 . . . . . . . . . . . 12 ⟨1, 0⟩ ∈ (ℝ × ℝ)
1514a1i 11 . . . . . . . . . . 11 ((𝜑𝑙𝑋) → ⟨1, 0⟩ ∈ (ℝ × ℝ))
16 eqid 2758 . . . . . . . . . . 11 (𝑙𝑋 ↦ ⟨1, 0⟩) = (𝑙𝑋 ↦ ⟨1, 0⟩)
1715, 16fmptd 6869 . . . . . . . . . 10 (𝜑 → (𝑙𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ))
18 reex 10666 . . . . . . . . . . . . 13 ℝ ∈ V
1918, 18xpex 7474 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝜑 → (ℝ × ℝ) ∈ V)
21 ovn0lem.x . . . . . . . . . . 11 (𝜑𝑋 ∈ Fin)
22 elmapg 8429 . . . . . . . . . . 11 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
2320, 21, 22syl2anc 587 . . . . . . . . . 10 (𝜑 → ((𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
2417, 23mpbird 260 . . . . . . . . 9 (𝜑 → (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
2524adantr 484 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
26 ovn0lem.i . . . . . . . 8 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨1, 0⟩))
2725, 26fmptd 6869 . . . . . . 7 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
28 ovexd 7185 . . . . . . . 8 (𝜑 → ((ℝ × ℝ) ↑m 𝑋) ∈ V)
29 nnex 11680 . . . . . . . . 9 ℕ ∈ V
3029a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
31 elmapg 8429 . . . . . . . 8 ((((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
3228, 30, 31syl2anc 587 . . . . . . 7 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
3327, 32mpbird 260 . . . . . 6 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
34 ovn0lem.n0 . . . . . . . . . . . 12 (𝜑𝑋 ≠ ∅)
35 n0 4245 . . . . . . . . . . . 12 (𝑋 ≠ ∅ ↔ ∃𝑙 𝑙𝑋)
3634, 35sylib 221 . . . . . . . . . . 11 (𝜑 → ∃𝑙 𝑙𝑋)
3736adantr 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∃𝑙 𝑙𝑋)
38 nfv 1915 . . . . . . . . . . . . 13 𝑘((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋)
39 nfcv 2919 . . . . . . . . . . . . 13 𝑘(vol‘(([,) ∘ (𝐼𝑗))‘𝑙))
4021ad2antrr 725 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → 𝑋 ∈ Fin)
4127ffvelrnda 6842 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
42 elmapi 8438 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
4443adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
45 simpr 488 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
4644, 45fvovco 42191 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
47 simpr 488 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
4825elexd 3430 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ V)
4926fvmpt2 6770 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑗 ∈ ℕ ∧ (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ V) → (𝐼𝑗) = (𝑙𝑋 ↦ ⟨1, 0⟩))
5047, 48, 49syl2anc 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) = (𝑙𝑋 ↦ ⟨1, 0⟩))
5150adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗) = (𝑙𝑋 ↦ ⟨1, 0⟩))
52 eqidd 2759 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → ⟨1, 0⟩ = ⟨1, 0⟩)
5314elexi 3429 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨1, 0⟩ ∈ V
5453a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨1, 0⟩ ∈ V)
5551, 52, 45, 54fvmptd 6766 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐼𝑗)‘𝑘) = ⟨1, 0⟩)
5655fveq2d 6662 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = (1st ‘⟨1, 0⟩))
5710elexi 3429 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
584elexi 3429 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
5957, 58op1st 7701 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨1, 0⟩) = 1
6059a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘⟨1, 0⟩) = 1)
6156, 60eqtrd 2793 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = 1)
6255fveq2d 6662 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = (2nd ‘⟨1, 0⟩))
6357, 58op2nd 7702 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨1, 0⟩) = 0
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘⟨1, 0⟩) = 0)
6562, 64eqtrd 2793 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = 0)
6661, 65oveq12d 7168 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (1[,)0))
67 0le1 11201 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 1
68 1xr 10738 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℝ*
69 ico0 12825 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0) = ∅ ↔ 0 ≤ 1))
7068, 4, 69mp2an 691 . . . . . . . . . . . . . . . . . . . 20 ((1[,)0) = ∅ ↔ 0 ≤ 1)
7167, 70mpbir 234 . . . . . . . . . . . . . . . . . . 19 (1[,)0) = ∅
7271a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1[,)0) = ∅)
7346, 66, 723eqtrd 2797 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ∅)
7473fveq2d 6662 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘∅))
75 vol0 42967 . . . . . . . . . . . . . . . . 17 (vol‘∅) = 0
7675a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘∅) = 0)
7774, 76eqtrd 2793 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0)
78 0cn 10671 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
7978a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 0 ∈ ℂ)
8077, 79eqeltrd 2852 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ ℂ)
8180adantlr 714 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ ℂ)
82 2fveq3 6663 . . . . . . . . . . . . 13 (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)))
83 simpr 488 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → 𝑙𝑋)
84 eleq1w 2834 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (𝑘𝑋𝑙𝑋))
8584anbi2d 631 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) ↔ ((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋)))
8682eqeq1d 2760 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)) = 0))
8785, 86imbi12d 348 . . . . . . . . . . . . . 14 (𝑘 = 𝑙 → ((((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0) ↔ (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)) = 0)))
8887, 77chvarvv 2005 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)) = 0)
8938, 39, 40, 81, 82, 83, 88fprod0 42604 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0)
9089ex 416 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑙𝑋 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0))
9190exlimdv 1934 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (∃𝑙 𝑙𝑋 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0))
9237, 91mpd 15 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0)
9392mpteq2dva 5127 . . . . . . . 8 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0))
9493fveq2d 6662 . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ 0)))
95 nfv 1915 . . . . . . . 8 𝑗𝜑
9695, 30sge0z 43380 . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 0)) = 0)
97 eqidd 2759 . . . . . . 7 (𝜑 → 0 = 0)
9894, 96, 973eqtrrd 2798 . . . . . 6 (𝜑 → 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
99 fveq1 6657 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
10099coeq2d 5702 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
101100fveq1d 6660 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
102101fveq2d 6662 . . . . . . . . . . 11 (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
103102ralrimivw 3114 . . . . . . . . . 10 (𝑖 = 𝐼 → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
104103prodeq2d 15324 . . . . . . . . 9 (𝑖 = 𝐼 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
105104mpteq2dv 5128 . . . . . . . 8 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
106105fveq2d 6662 . . . . . . 7 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
107106rspceeqv 3556 . . . . . 6 ((𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
10833, 98, 107syl2anc 587 . . . . 5 (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
1095, 108jca 515 . . . 4 (𝜑 → (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
110 eqeq1 2762 . . . . . 6 (𝑧 = 0 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
111110rexbidv 3221 . . . . 5 (𝑧 = 0 → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
112111, 6elrab2 3605 . . . 4 (0 ∈ 𝑀 ↔ (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
113109, 112sylibr 237 . . 3 (𝜑 → 0 ∈ 𝑀)
114 infxrlb 12768 . . 3 ((𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 0)
1159, 113, 114syl2anc 587 . 2 (𝜑 → inf(𝑀, ℝ*, < ) ≤ 0)
116 pnfxr 10733 . . . 4 +∞ ∈ ℝ*
117116a1i 11 . . 3 (𝜑 → +∞ ∈ ℝ*)
118 iccgelb 12835 . . 3 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, < ))
1195, 117, 2, 118syl3anc 1368 . 2 (𝜑 → 0 ≤ inf(𝑀, ℝ*, < ))
1203, 5, 115, 119xrletrid 12589 1 (𝜑 → inf(𝑀, ℝ*, < ) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2951  ∃wrex 3071  {crab 3074  Vcvv 3409   ⊆ wss 3858  ∅c0 4225  ⟨cop 4528   class class class wbr 5032   ↦ cmpt 5112   × cxp 5522   ∘ ccom 5528  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692   ↑m cmap 8416  Fincfn 8527  infcinf 8938  ℂcc 10573  ℝcr 10574  0cc0 10575  1c1 10576  +∞cpnf 10710  ℝ*cxr 10712   < clt 10713   ≤ cle 10714  ℕcn 11674  [,)cico 12781  [,]cicc 12782  ∏cprod 15307  volcvol 24163  Σ^csumge0 43367 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-inf2 9137  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652  ax-pre-sup 10653 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 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7405  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-2o 8113  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-sup 8939  df-inf 8940  df-oi 9007  df-dju 9363  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-div 11336  df-nn 11675  df-2 11737  df-3 11738  df-n0 11935  df-z 12021  df-uz 12283  df-q 12389  df-rp 12431  df-xadd 12549  df-ioo 12783  df-ico 12785  df-icc 12786  df-fz 12940  df-fzo 13083  df-fl 13211  df-seq 13419  df-exp 13480  df-hash 13741  df-cj 14506  df-re 14507  df-im 14508  df-sqrt 14642  df-abs 14643  df-clim 14893  df-sum 15091  df-prod 15308  df-xmet 20159  df-met 20160  df-ovol 24164  df-vol 24165  df-sumge0 43368 This theorem is referenced by:  ovn0  43571
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