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Theorem ovn0lem 42715
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 12812 . . 3 (0[,]+∞) ⊆ ℝ*
2 ovn0lem.infm . . 3 (𝜑 → inf(𝑀, ℝ*, < ) ∈ (0[,]+∞))
31, 2sseldi 3968 . 2 (𝜑 → inf(𝑀, ℝ*, < ) ∈ ℝ*)
4 0xr 10680 . . 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 4059 . . . . 5 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))} ⊆ ℝ*
86, 7eqsstri 4004 . . . 4 𝑀 ⊆ ℝ*
98a1i 11 . . 3 (𝜑𝑀 ⊆ ℝ*)
10 1re 10633 . . . . . . . . . . . . . 14 1 ∈ ℝ
11 0re 10635 . . . . . . . . . . . . . 14 0 ∈ ℝ
1210, 11pm3.2i 471 . . . . . . . . . . . . 13 (1 ∈ ℝ ∧ 0 ∈ ℝ)
13 opelxp 5589 . . . . . . . . . . . . 13 (⟨1, 0⟩ ∈ (ℝ × ℝ) ↔ (1 ∈ ℝ ∧ 0 ∈ ℝ))
1412, 13mpbir 232 . . . . . . . . . . . 12 ⟨1, 0⟩ ∈ (ℝ × ℝ)
1514a1i 11 . . . . . . . . . . 11 ((𝜑𝑙𝑋) → ⟨1, 0⟩ ∈ (ℝ × ℝ))
16 eqid 2825 . . . . . . . . . . 11 (𝑙𝑋 ↦ ⟨1, 0⟩) = (𝑙𝑋 ↦ ⟨1, 0⟩)
1715, 16fmptd 6873 . . . . . . . . . 10 (𝜑 → (𝑙𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ))
18 reex 10620 . . . . . . . . . . . . 13 ℝ ∈ V
1918, 18xpex 7468 . . . . . . . . . . . 12 (ℝ × ℝ) ∈ V
2019a1i 11 . . . . . . . . . . 11 (𝜑 → (ℝ × ℝ) ∈ V)
21 ovn0lem.x . . . . . . . . . . 11 (𝜑𝑋 ∈ Fin)
22 elmapg 8412 . . . . . . . . . . 11 (((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
2320, 21, 22syl2anc 584 . . . . . . . . . 10 (𝜑 → ((𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋) ↔ (𝑙𝑋 ↦ ⟨1, 0⟩):𝑋⟶(ℝ × ℝ)))
2417, 23mpbird 258 . . . . . . . . 9 (𝜑 → (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
2524adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ ((ℝ × ℝ) ↑m 𝑋))
26 ovn0lem.i . . . . . . . 8 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙𝑋 ↦ ⟨1, 0⟩))
2725, 26fmptd 6873 . . . . . . 7 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
28 ovexd 7186 . . . . . . . 8 (𝜑 → ((ℝ × ℝ) ↑m 𝑋) ∈ V)
29 nnex 11636 . . . . . . . . 9 ℕ ∈ V
3029a1i 11 . . . . . . . 8 (𝜑 → ℕ ∈ V)
31 elmapg 8412 . . . . . . . 8 ((((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
3228, 30, 31syl2anc 584 . . . . . . 7 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
3327, 32mpbird 258 . . . . . 6 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
34 ovn0lem.n0 . . . . . . . . . . . 12 (𝜑𝑋 ≠ ∅)
35 n0 4313 . . . . . . . . . . . 12 (𝑋 ≠ ∅ ↔ ∃𝑙 𝑙𝑋)
3634, 35sylib 219 . . . . . . . . . . 11 (𝜑 → ∃𝑙 𝑙𝑋)
3736adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∃𝑙 𝑙𝑋)
38 nfv 1908 . . . . . . . . . . . . 13 𝑘((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋)
39 nfcv 2981 . . . . . . . . . . . . 13 𝑘(vol‘(([,) ∘ (𝐼𝑗))‘𝑙))
4021ad2antrr 722 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → 𝑋 ∈ Fin)
4127ffvelrnda 6846 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
42 elmapi 8421 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
4341, 42syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
4443adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
45 simpr 485 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
4644, 45fvovco 41322 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
47 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
4825elexd 3519 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ ℕ) → (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ V)
4926fvmpt2 6774 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑗 ∈ ℕ ∧ (𝑙𝑋 ↦ ⟨1, 0⟩) ∈ V) → (𝐼𝑗) = (𝑙𝑋 ↦ ⟨1, 0⟩))
5047, 48, 49syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) = (𝑙𝑋 ↦ ⟨1, 0⟩))
5150adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗) = (𝑙𝑋 ↦ ⟨1, 0⟩))
52 eqidd 2826 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) ∧ 𝑙 = 𝑘) → ⟨1, 0⟩ = ⟨1, 0⟩)
5314elexi 3518 . . . . . . . . . . . . . . . . . . . . . . 23 ⟨1, 0⟩ ∈ V
5453a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ⟨1, 0⟩ ∈ V)
5551, 52, 45, 54fvmptd 6770 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐼𝑗)‘𝑘) = ⟨1, 0⟩)
5655fveq2d 6670 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = (1st ‘⟨1, 0⟩))
5710elexi 3518 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ V
584elexi 3518 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ V
5957, 58op1st 7691 . . . . . . . . . . . . . . . . . . . . 21 (1st ‘⟨1, 0⟩) = 1
6059a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘⟨1, 0⟩) = 1)
6156, 60eqtrd 2860 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = 1)
6255fveq2d 6670 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = (2nd ‘⟨1, 0⟩))
6357, 58op2nd 7692 . . . . . . . . . . . . . . . . . . . . 21 (2nd ‘⟨1, 0⟩) = 0
6463a1i 11 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘⟨1, 0⟩) = 0)
6562, 64eqtrd 2860 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = 0)
6661, 65oveq12d 7169 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (1[,)0))
67 0le1 11155 . . . . . . . . . . . . . . . . . . . 20 0 ≤ 1
68 1xr 10692 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℝ*
69 ico0 12777 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0) = ∅ ↔ 0 ≤ 1))
7068, 4, 69mp2an 688 . . . . . . . . . . . . . . . . . . . 20 ((1[,)0) = ∅ ↔ 0 ≤ 1)
7167, 70mpbir 232 . . . . . . . . . . . . . . . . . . 19 (1[,)0) = ∅
7271a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1[,)0) = ∅)
7346, 66, 723eqtrd 2864 . . . . . . . . . . . . . . . . 17 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ∅)
7473fveq2d 6670 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘∅))
75 vol0 42111 . . . . . . . . . . . . . . . . 17 (vol‘∅) = 0
7675a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘∅) = 0)
7774, 76eqtrd 2860 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0)
78 0cn 10625 . . . . . . . . . . . . . . . 16 0 ∈ ℂ
7978a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 0 ∈ ℂ)
8077, 79eqeltrd 2917 . . . . . . . . . . . . . 14 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ ℂ)
8180adantlr 711 . . . . . . . . . . . . 13 ((((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ ℂ)
82 2fveq3 6671 . . . . . . . . . . . . 13 (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)))
83 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → 𝑙𝑋)
84 eleq1w 2899 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑙 → (𝑘𝑋𝑙𝑋))
8584anbi2d 628 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) ↔ ((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋)))
8682eqeq1d 2827 . . . . . . . . . . . . . . 15 (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)) = 0))
8785, 86imbi12d 346 . . . . . . . . . . . . . 14 (𝑘 = 𝑙 → ((((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0) ↔ (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)) = 0)))
8887, 77chvarv 2410 . . . . . . . . . . . . 13 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → (vol‘(([,) ∘ (𝐼𝑗))‘𝑙)) = 0)
8938, 39, 40, 81, 82, 83, 88fprod0 41744 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ ℕ) ∧ 𝑙𝑋) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0)
9089ex 413 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑙𝑋 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0))
9190exlimdv 1927 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (∃𝑙 𝑙𝑋 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0))
9237, 91mpd 15 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) = 0)
9392mpteq2dva 5157 . . . . . . . 8 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0))
9493fveq2d 6670 . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ 0)))
95 nfv 1908 . . . . . . . 8 𝑗𝜑
9695, 30sge0z 42525 . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ 0)) = 0)
97 eqidd 2826 . . . . . . 7 (𝜑 → 0 = 0)
9894, 96, 973eqtrrd 2865 . . . . . 6 (𝜑 → 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
99 fveq1 6665 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
10099coeq2d 5731 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
101100fveq1d 6668 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
102101fveq2d 6670 . . . . . . . . . . 11 (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
103102ralrimivw 3187 . . . . . . . . . 10 (𝑖 = 𝐼 → ∀𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
104103prodeq2d 15268 . . . . . . . . 9 (𝑖 = 𝐼 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
105104mpteq2dv 5158 . . . . . . . 8 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
106105fveq2d 6670 . . . . . . 7 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
107106rspceeqv 3641 . . . . . 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 584 . . . . 5 (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))
1095, 108jca 512 . . . 4 (𝜑 → (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
110 eqeq1 2829 . . . . . 6 (𝑧 = 0 → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ 0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
111110rexbidv 3301 . . . . 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 3686 . . . 4 (0 ∈ 𝑀 ↔ (0 ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
113109, 112sylibr 235 . . 3 (𝜑 → 0 ∈ 𝑀)
114 infxrlb 12720 . . 3 ((𝑀 ⊆ ℝ* ∧ 0 ∈ 𝑀) → inf(𝑀, ℝ*, < ) ≤ 0)
1159, 113, 114syl2anc 584 . 2 (𝜑 → inf(𝑀, ℝ*, < ) ≤ 0)
116 pnfxr 10687 . . . 4 +∞ ∈ ℝ*
117116a1i 11 . . 3 (𝜑 → +∞ ∈ ℝ*)
118 iccgelb 12786 . . 3 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ inf(𝑀, ℝ*, < ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, < ))
1195, 117, 2, 118syl3anc 1365 . 2 (𝜑 → 0 ≤ inf(𝑀, ℝ*, < ))
1203, 5, 115, 119xrletrid 12541 1 (𝜑 → inf(𝑀, ℝ*, < ) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wne 3020  wrex 3143  {crab 3146  Vcvv 3499  wss 3939  c0 4294  cop 4569   class class class wbr 5062  cmpt 5142   × cxp 5551  ccom 5557  wf 6347  cfv 6351  (class class class)co 7151  1st c1st 7681  2nd c2nd 7682  m cmap 8399  Fincfn 8501  infcinf 8897  cc 10527  cr 10528  0cc0 10529  1c1 10530  +∞cpnf 10664  *cxr 10666   < clt 10667  cle 10668  cn 11630  [,)cico 12733  [,]cicc 12734  cprod 15251  volcvol 23981  Σ^csumge0 42512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-of 7402  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-q 12341  df-rp 12383  df-xadd 12501  df-ioo 12735  df-ico 12737  df-icc 12738  df-fz 12886  df-fzo 13027  df-fl 13155  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-sum 15036  df-prod 15252  df-xmet 20456  df-met 20457  df-ovol 23982  df-vol 23983  df-sumge0 42513
This theorem is referenced by:  ovn0  42716
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