Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovnlecvr Structured version   Visualization version   GIF version

Theorem ovnlecvr 41270
 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 6514 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
6 elmapi 8037 . . . . . . . 8 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
75, 6syl 17 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
87hoissrrn 41261 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
98ralrimiva 3096 . . . . 5 (𝜑 → ∀𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
10 iunss 4705 . . . . 5 ( 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋) ↔ ∀𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
119, 10sylibr 224 . . . 4 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ⊆ (ℝ ↑𝑚 𝑋))
123, 11sstrd 3746 . . 3 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
13 eqid 2752 . . 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 41263 . 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 3820 . . . 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 11210 . . . . . . 7 ℕ ∈ V
1817a1i 11 . . . . . 6 (𝜑 → ℕ ∈ V)
19 icossicc 12445 . . . . . . . 8 (0[,)+∞) ⊆ (0[,]+∞)
20 nfv 1984 . . . . . . . . 9 𝑘(𝜑𝑗 ∈ ℕ)
211adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
22 ovnlecvr.l . . . . . . . . 9 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)))
2320, 21, 22, 7hoiprodcl2 41267 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) ∈ (0[,)+∞))
2419, 23sseldi 3734 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) ∈ (0[,]+∞))
25 eqid 2752 . . . . . . 7 (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))
2624, 25fmptd 6540 . . . . . 6 (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))):ℕ⟶(0[,]+∞))
2718, 26sge0xrcl 41097 . . . . 5 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ∈ ℝ*)
28 ovex 6833 . . . . . . . . 9 ((ℝ × ℝ) ↑𝑚 𝑋) ∈ V
2928, 17pm3.2i 470 . . . . . . . 8 (((ℝ × ℝ) ↑𝑚 𝑋) ∈ V ∧ ℕ ∈ V)
30 elmapg 8028 . . . . . . . 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 224 . . . . . 6 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
3322a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → 𝐿 = (𝑖 ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘))))
34 coeq2 5428 . . . . . . . . . . . . . 14 (𝑖 = (𝐼𝑗) → ([,) ∘ 𝑖) = ([,) ∘ (𝐼𝑗)))
3534fveq1d 6346 . . . . . . . . . . . . 13 (𝑖 = (𝐼𝑗) → (([,) ∘ 𝑖)‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
3635fveq2d 6348 . . . . . . . . . . . 12 (𝑖 = (𝐼𝑗) → (vol‘(([,) ∘ 𝑖)‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
3736prodeq2ad 40319 . . . . . . . . . . 11 (𝑖 = (𝐼𝑗) → ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
3837adantl 473 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑖 = (𝐼𝑗)) → ∏𝑘𝑋 (vol‘(([,) ∘ 𝑖)‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
39 prodex 14828 . . . . . . . . . . 11 𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)) ∈ V)
4133, 38, 5, 40fvmptd 6442 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
4241mpteq2dva 4888 . . . . . . . 8 (𝜑 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
4342fveq2d 6348 . . . . . . 7 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
443, 43jca 555 . . . . . 6 (𝜑 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))))
45 nfv 1984 . . . . . . . . . . 11 𝑘 𝑖 = 𝐼
46 fveq1 6343 . . . . . . . . . . . . . 14 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4746coeq2d 5432 . . . . . . . . . . . . 13 (𝑖 = 𝐼 → ([,) ∘ (𝑖𝑗)) = ([,) ∘ (𝐼𝑗)))
4847fveq1d 6346 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
4948adantr 472 . . . . . . . . . . 11 ((𝑖 = 𝐼𝑘𝑋) → (([,) ∘ (𝑖𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
5045, 49ixpeq2d 39728 . . . . . . . . . 10 (𝑖 = 𝐼X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
5150iuneq2d 4691 . . . . . . . . 9 (𝑖 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
5251sseq2d 3766 . . . . . . . 8 (𝑖 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑖𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
5348fveq2d 6348 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
5453prodeq2ad 40319 . . . . . . . . . . 11 (𝑖 = 𝐼 → ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)) = ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))
5554mpteq2dv 4889 . . . . . . . . . 10 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘))))
5655fveq2d 6348 . . . . . . . . 9 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝑖𝑗))‘𝑘)))) = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(([,) ∘ (𝐼𝑗))‘𝑘)))))
5756eqeq2d 2762 . . . . . . . 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))))))
5852, 57anbi12d 749 . . . . . . 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)))))))
5958rspcev 3441 . . . . . 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))))))
6032, 44, 59syl2anc 696 . . . . 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))))))
6127, 60jca 555 . . . 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)))))))
62 eqeq1 2756 . . . . . . 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))))))
6362anbi2d 742 . . . . . 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)))))))
6463rexbidv 3182 . . . . 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)))))))
6564elrab 3496 . . . 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)))))))
6661, 65sylibr 224 . . 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)))))})
67 infxrlb 12349 . . 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)))))
6816, 66, 67syl2anc 696 . 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)))))
6914, 68eqbrtrd 4818 1 (𝜑 → ((voln*‘𝑋)‘𝐴) ≤ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  ∀wral 3042  ∃wrex 3043  {crab 3046  Vcvv 3332   ⊆ wss 3707  ∅c0 4050  ∪ ciun 4664   class class class wbr 4796   ↦ cmpt 4873   × cxp 5256   ∘ ccom 5262  ⟶wf 6037  ‘cfv 6041  (class class class)co 6805   ↑𝑚 cmap 8015  Xcixp 8066  Fincfn 8113  infcinf 8504  ℝcr 10119  0cc0 10120  +∞cpnf 10255  ℝ*cxr 10257   < clt 10258   ≤ cle 10259  ℕcn 11204  [,)cico 12362  [,]cicc 12363  ∏cprod 14826  volcvol 23424  Σ^csumge0 41074  voln*covoln 41248 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197  ax-pre-sup 10198 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-fal 1630  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-se 5218  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-isom 6050  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-pm 8018  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8474  df-sup 8505  df-inf 8506  df-oi 8572  df-card 8947  df-cda 9174  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-nn 11205  df-2 11263  df-3 11264  df-n0 11477  df-z 11562  df-uz 11872  df-q 11974  df-rp 12018  df-xneg 12131  df-xadd 12132  df-xmul 12133  df-ioo 12364  df-ico 12366  df-icc 12367  df-fz 12512  df-fzo 12652  df-fl 12779  df-seq 12988  df-exp 13047  df-hash 13304  df-cj 14030  df-re 14031  df-im 14032  df-sqrt 14166  df-abs 14167  df-clim 14410  df-rlim 14411  df-sum 14608  df-prod 14827  df-rest 16277  df-topgen 16298  df-psmet 19932  df-xmet 19933  df-met 19934  df-bl 19935  df-mopn 19936  df-top 20893  df-topon 20910  df-bases 20944  df-cmp 21384  df-ovol 23425  df-vol 23426  df-sumge0 41075  df-ovoln 41249 This theorem is referenced by:  ovnsubaddlem1  41282
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