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Theorem ovnlecvr 42717
Description: Given a subset of multidimensional reals and a set of half-open intervals that covers it, the Lebesgue outer measure of the set is bounded by the generalized sum of the pre-measure of the half-open intervals. The statement would also be true with 𝑋 the empty set, but covers are not used for the zero-dimensional case. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
ovnlecvr.x (𝜑𝑋 ∈ Fin)
ovnlecvr.n0 (𝜑𝑋 ≠ ∅)
ovnlecvr.l 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
ovnlecvr.i (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
ovnlecvr.ss (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
Assertion
Ref Expression
ovnlecvr (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
Distinct variable groups:   𝐴,𝑖   𝑖,𝐼,𝑗,𝑘   𝑖,𝐿   𝑖,𝑋,𝑗,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐿(𝑗,𝑘)

Proof of Theorem ovnlecvr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ovnlecvr.x . . 3 (𝜑𝑋 ∈ Fin)
2 ovnlecvr.n0 . . 3 (𝜑𝑋 ≠ ∅)
3 ovnlecvr.ss . . . 4 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
4 ovnlecvr.i . . . . . . . . 9 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
54ffvelrnda 6843 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
6 elmapi 8417 . . . . . . . 8 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
75, 6syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
87hoissrrn 42708 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
98ralrimiva 3179 . . . . 5 (𝜑 → ∀𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
10 iunss 4960 . . . . 5 ( 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋) ↔ ∀𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
119, 10sylibr 235 . . . 4 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑m 𝑋))
123, 11sstrd 3974 . . 3 (𝜑𝐴 ⊆ (ℝ ↑m 𝑋))
13 eqid 2818 . . 3 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}
141, 2, 12, 13ovnn0val 42710 . 2 (𝜑 → ((voln*‘𝑋)‘𝐴) = inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ))
15 ssrab2 4053 . . . 4 {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ*
1615a1i 11 . . 3 (𝜑 → {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ*)
17 nnex 11632 . . . . . . 7 ℕ ∈ V
1817a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
19 icossicc 12812 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
20 nfv 1906 . . . . . . . . 9 𝑘(𝜑𝑗 ∈ ℕ)
211adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
22 ovnlecvr.l . . . . . . . . 9 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑m 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
2320, 21, 22, 7hoiprodcl2 42714 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) ∈ (0[,)+∞))
2419, 23sseldi 3962 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) ∈ (0[,]+∞))
25 eqid 2818 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))
2624, 25fmptd 6870 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))):ℕ⟶(0[,]+∞))
2718, 26sge0xrcl 42544 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ ℝ*)
28 ovex 7178 . . . . . . . . 9 ((ℝ × ℝ) ↑m 𝑋) ∈ V
2928, 17pm3.2i 471 . . . . . . . 8 (((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V)
30 elmapg 8408 . . . . . . . 8 ((((ℝ × ℝ) ↑m 𝑋) ∈ V ∧ ℕ ∈ V) → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋)))
3129, 30ax-mp 5 . . . . . . 7 (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ↔ 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
324, 31sylibr 235 . . . . . 6 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
33 coeq2 5722 . . . . . . . . . . . . 13 (𝑖 = (𝐼𝑗) → ([,) ∘ 𝑖) = ([,) ∘ (𝐼𝑗)))
3433fveq1d 6665 . . . . . . . . . . . 12 (𝑖 = (𝐼𝑗) → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
3534fveq2d 6667 . . . . . . . . . . 11 (𝑖 = (𝐼𝑗) → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
3635prodeq2ad 41749 . . . . . . . . . 10 (𝑖 = (𝐼𝑗) → ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
37 prodex 15249 . . . . . . . . . . 11 𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ V
3837a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ V)
3922, 36, 5, 38fvmptd3 6783 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
4039mpteq2dva 5152 . . . . . . . 8 (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
4140fveq2d 6667 . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
423, 41jca 512 . . . . . 6 (𝜑 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
43 nfv 1906 . . . . . . . . . . 11 𝑘 𝑖 = 𝐼
44 fveq1 6662 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4544coeq2d 5726 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
4645fveq1d 6665 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
4746adantr 481 . . . . . . . . . . 11 ((𝑖 = 𝐼𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
4843, 47ixpeq2d 41207 . . . . . . . . . 10 (𝑖 = 𝐼X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
4948iuneq2d 4939 . . . . . . . . 9 (𝑖 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
5049sseq2d 3996 . . . . . . . 8 (𝑖 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
5146fveq2d 6667 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
5251prodeq2ad 41749 . . . . . . . . . . 11 (𝑖 = 𝐼 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
5352mpteq2dv 5153 . . . . . . . . . 10 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
5453fveq2d 6667 . . . . . . . . 9 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
5554eqeq2d 2829 . . . . . . . 8 (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
5650, 55anbi12d 630 . . . . . . 7 (𝑖 = 𝐼 → ((𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))))
5756rspcev 3620 . . . . . 6 ((𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))) → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
5832, 42, 57syl2anc 584 . . . . 5 (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
5927, 58jca 512 . . . 4 (𝜑 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
60 eqeq1 2822 . . . . . . 7 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) → (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))))
6160anbi2d 628 . . . . . 6 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) → ((𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
6261rexbidv 3294 . . . . 5 (𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) → (∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
6362elrab 3677 . . . 4 ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ↔ ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))))
6459, 63sylibr 235 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))})
65 infxrlb 12715 . . 3 (({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))} ⊆ ℝ* ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ {𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}) → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
6616, 64, 65syl2anc 584 . 2 (𝜑 → inf({𝑧 ∈ ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ∧ 𝑧 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))))}, ℝ*, < ) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
6714, 66eqbrtrd 5079 1 (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  c0 4288   ciun 4910   class class class wbr 5057  cmpt 5137   × cxp 5546  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395  Xcixp 8449  Fincfn 8497  infcinf 8893  cr 10524  0cc0 10525  +∞cpnf 10660  *cxr 10662   < clt 10663  cle 10664  cn 11626  [,)cico 12728  [,]cicc 12729  cprod 15247  volcvol 23991  Σ^csumge0 42521  voln*covoln 42695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-seq 13358  df-exp 13418  df-hash 13679  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-clim 14833  df-rlim 14834  df-sum 15031  df-prod 15248  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923  df-ovol 23992  df-vol 23993  df-sumge0 42522  df-ovoln 42696
This theorem is referenced by:  ovnsubaddlem1  42729
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