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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnlecvr Structured version   Visualization version   GIF version

Theorem ovnlecvr 42402
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 6716 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
6 elmapi 8278 . . . . . . . 8 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
75, 6syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
87hoissrrn 42393 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
98ralrimiva 3149 . . . . 5 (𝜑 → ∀𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
10 iunss 4868 . . . . 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 3899 . . 3 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
13 eqid 2795 . . 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 42395 . 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 3977 . . . 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 11492 . . . . . . 7 ℕ ∈ V
1817a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
19 icossicc 12674 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
20 nfv 1892 . . . . . . . . 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 42399 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) ∈ (0[,)+∞))
2419, 23sseldi 3887 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) ∈ (0[,]+∞))
25 eqid 2795 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))
2624, 25fmptd 6741 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))):ℕ⟶(0[,]+∞))
2718, 26sge0xrcl 42229 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ ℝ*)
28 ovex 7048 . . . . . . . . 9 ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
2928, 17pm3.2i 471 . . . . . . . 8 (((ℝ × ℝ) ↑𝑚 𝑋) ∈ V ∧ ℕ ∈ V)
30 elmapg 8269 . . . . . . . 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 5615 . . . . . . . . . . . . 13 (𝑖 = (𝐼𝑗) → ([,) ∘ 𝑖) = ([,) ∘ (𝐼𝑗)))
3433fveq1d 6540 . . . . . . . . . . . 12 (𝑖 = (𝐼𝑗) → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
3534fveq2d 6542 . . . . . . . . . . 11 (𝑖 = (𝐼𝑗) → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
3635prodeq2ad 41434 . . . . . . . . . 10 (𝑖 = (𝐼𝑗) → ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
37 prodex 15094 . . . . . . . . . . 11 𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ V
3837a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ V)
3922, 36, 5, 38fvmptd3 6657 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
4039mpteq2dva 5055 . . . . . . . 8 (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
4140fveq2d 6542 . . . . . . 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 1892 . . . . . . . . . . 11 𝑘 𝑖 = 𝐼
44 fveq1 6537 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4544coeq2d 5619 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
4645fveq1d 6540 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
4746adantr 481 . . . . . . . . . . 11 ((𝑖 = 𝐼𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
4843, 47ixpeq2d 40888 . . . . . . . . . 10 (𝑖 = 𝐼X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
4948iuneq2d 4853 . . . . . . . . 9 (𝑖 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
5049sseq2d 3920 . . . . . . . 8 (𝑖 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
5146fveq2d 6542 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
5251prodeq2ad 41434 . . . . . . . . . . 11 (𝑖 = 𝐼 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
5352mpteq2dv 5056 . . . . . . . . . 10 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
5453fveq2d 6542 . . . . . . . . 9 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
5554eqeq2d 2805 . . . . . . . 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 3559 . . . . . 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 2799 . . . . . . 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 3260 . . . . 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 3618 . . . 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 12577 . . 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 4984 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 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  wss 3859  c0 4211   ciun 4825   class class class wbr 4962  cmpt 5041   × cxp 5441  ccom 5447  wf 6221  cfv 6225  (class class class)co 7016  𝑚 cmap 8256  Xcixp 8310  Fincfn 8357  infcinf 8751  cr 10382  0cc0 10383  +∞cpnf 10518  *cxr 10520   < clt 10521  cle 10522  cn 11486  [,)cico 12590  [,]cicc 12591  cprod 15092  volcvol 23747  Σ^csumge0 42206  voln*covoln 42380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-pm 8259  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fi 8721  df-sup 8752  df-inf 8753  df-oi 8820  df-dju 9176  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-n0 11746  df-z 11830  df-uz 12094  df-q 12198  df-rp 12240  df-xneg 12357  df-xadd 12358  df-xmul 12359  df-ioo 12592  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-fl 13012  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-rlim 14680  df-sum 14877  df-prod 15093  df-rest 16525  df-topgen 16546  df-psmet 20219  df-xmet 20220  df-met 20221  df-bl 20222  df-mopn 20223  df-top 21186  df-topon 21203  df-bases 21238  df-cmp 21679  df-ovol 23748  df-vol 23749  df-sumge0 42207  df-ovoln 42381
This theorem is referenced by:  ovnsubaddlem1  42414
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