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Theorem ovncvr2 42330
Description: 𝐵 and 𝑇 are the left and right side of a cover of 𝐴. This cover is made of n-dimensional half-open intervals and approximates the n-dimensional Lebesgue outer volume of 𝐴. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
ovncvr2.x (𝜑𝑋 ∈ Fin)
ovncvr2.a (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
ovncvr2.e (𝜑𝐸 ∈ ℝ+)
ovncvr2.c 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
ovncvr2.l 𝐿 = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
ovncvr2.d 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))
ovncvr2.i (𝜑𝐼 ∈ ((𝐷𝐴)‘𝐸))
ovncvr2.b 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
ovncvr2.t 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
Assertion
Ref Expression
ovncvr2 (𝜑 → (((𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
Distinct variable groups:   𝐴,𝑎,𝑖,𝑟   𝐴,𝑙,𝑎   𝐵,   𝐶,𝑎,𝑖,𝑟   𝑖,𝐸,𝑟   ,𝐼,𝑗,𝑘   𝑖,𝐼,𝑗   𝐼,𝑙,𝑗,𝑘   𝐿,𝑎,𝑖,𝑟   𝑇,   𝑋,𝑎,𝑖,𝑗,𝑟   ,𝑋,𝑘   𝑋,𝑙   𝑘,𝑎,𝜑,𝑗   𝜑,   𝜑,𝑟
Allowed substitution hints:   𝜑(𝑖,𝑙)   𝐴(,𝑗,𝑘)   𝐵(𝑖,𝑗,𝑘,𝑟,𝑎,𝑙)   𝐶(,𝑗,𝑘,𝑙)   𝐷(,𝑖,𝑗,𝑘,𝑟,𝑎,𝑙)   𝑇(𝑖,𝑗,𝑘,𝑟,𝑎,𝑙)   𝐸(,𝑗,𝑘,𝑎,𝑙)   𝐼(𝑟,𝑎)   𝐿(,𝑗,𝑘,𝑙)

Proof of Theorem ovncvr2
StepHypRef Expression
1 ovncvr2.c . . . . . . . . . . . . . . . 16 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
2 sseq1 3882 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐴 → (𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
32rabbidv 3403 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐴 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
4 ovncvr2.a . . . . . . . . . . . . . . . . 17 (𝜑𝐴 ⊆ (ℝ ↑𝑚 𝑋))
5 ovexd 7010 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ↑𝑚 𝑋) ∈ V)
65, 4ssexd 5084 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
7 elpwg 4430 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↔ 𝐴 ⊆ (ℝ ↑𝑚 𝑋)))
86, 7syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↔ 𝐴 ⊆ (ℝ ↑𝑚 𝑋)))
94, 8mpbird 249 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ 𝒫 (ℝ ↑𝑚 𝑋))
10 ovex 7008 . . . . . . . . . . . . . . . . . 18 (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∈ V
1110rabex 5091 . . . . . . . . . . . . . . . . 17 {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ∈ V
1211a1i 11 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ∈ V)
131, 3, 9, 12fvmptd3 6617 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
14 ssrab2 3946 . . . . . . . . . . . . . . . 16 {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)
1514a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ⊆ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
1613, 15eqsstrd 3895 . . . . . . . . . . . . . 14 (𝜑 → (𝐶𝐴) ⊆ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
17 ovncvr2.i . . . . . . . . . . . . . . . . 17 (𝜑𝐼 ∈ ((𝐷𝐴)‘𝐸))
18 ovncvr2.d . . . . . . . . . . . . . . . . . . 19 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑𝑚 𝑋) ↦ (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}))
19 fveq2 6499 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (𝐶𝑎) = (𝐶𝐴))
2019eleq2d 2851 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝐴 → (𝑖 ∈ (𝐶𝑎) ↔ 𝑖 ∈ (𝐶𝐴)))
21 fveq2 6499 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴))
2221oveq1d 6991 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))
2322breq2d 4941 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝐴 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)))
2420, 23anbi12d 621 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = 𝐴 → ((𝑖 ∈ (𝐶𝑎) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)) ↔ (𝑖 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))))
2524rabbidva2 3400 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = 𝐴 → {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)} = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)})
2625mpteq2dv 5023 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐴 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝑎) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
27 rpex 41049 . . . . . . . . . . . . . . . . . . . . 21 + ∈ V
2827mptex 6812 . . . . . . . . . . . . . . . . . . . 20 (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V
2928a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V)
3018, 26, 9, 29fvmptd3 6617 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
31 oveq2 6984 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
3231breq2d 4941 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
3332rabbidv 3403 . . . . . . . . . . . . . . . . . . 19 (𝑟 = 𝐸 → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)} = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
3433adantl 474 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑟 = 𝐸) → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)} = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
35 ovncvr2.e . . . . . . . . . . . . . . . . . 18 (𝜑𝐸 ∈ ℝ+)
36 fvex 6512 . . . . . . . . . . . . . . . . . . . 20 (𝐶𝐴) ∈ V
3736rabex 5091 . . . . . . . . . . . . . . . . . . 19 {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V
3837a1i 11 . . . . . . . . . . . . . . . . . 18 (𝜑 → {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V)
3930, 34, 35, 38fvmptd 6601 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐷𝐴)‘𝐸) = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
4017, 39eleqtrd 2868 . . . . . . . . . . . . . . . 16 (𝜑𝐼 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
41 fveq1 6498 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4241fveq2d 6503 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝐿‘(𝑖𝑗)) = (𝐿‘(𝐼𝑗)))
4342mpteq2dv 5023 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
4443fveq2d 6503 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
4544breq1d 4939 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4645elrab 3595 . . . . . . . . . . . . . . . 16 (𝐼 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ↔ (𝐼 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4740, 46sylib 210 . . . . . . . . . . . . . . 15 (𝜑 → (𝐼 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4847simpld 487 . . . . . . . . . . . . . 14 (𝜑𝐼 ∈ (𝐶𝐴))
4916, 48sseldd 3859 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ))
50 elmapi 8228 . . . . . . . . . . . . 13 (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
5149, 50syl 17 . . . . . . . . . . . 12 (𝜑𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
5251adantr 473 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑𝑚 𝑋))
53 simpr 477 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
5452, 53ffvelrnd 6677 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋))
55 elmapi 8228 . . . . . . . . . 10 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑𝑚 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5654, 55syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5756ffvelrnda 6676 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ))
58 xp1st 7533 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
5957, 58syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
6059fmpttd 6702 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
61 reex 10426 . . . . . . . . 9 ℝ ∈ V
6261a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
63 ovncvr2.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
64 elmapg 8219 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6562, 63, 64syl2anc 576 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6665adantr 473 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6760, 66mpbird 249 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋))
6867fmpttd 6702 . . . 4 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋))
69 ovncvr2.b . . . . . 6 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
7069a1i 11 . . . . 5 (𝜑𝐵 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘)))))
7170feq1d 6329 . . . 4 (𝜑 → (𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋)))
7268, 71mpbird 249 . . 3 (𝜑𝐵:ℕ⟶(ℝ ↑𝑚 𝑋))
73 xp2nd 7534 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7457, 73syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7574fmpttd 6702 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
76 elmapg 8219 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7762, 63, 76syl2anc 576 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7877adantr 473 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7975, 78mpbird 249 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑𝑚 𝑋))
8079fmpttd 6702 . . . 4 (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋))
81 ovncvr2.t . . . . . 6 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
8281a1i 11 . . . . 5 (𝜑𝑇 = (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘)))))
8382feq1d 6329 . . . 4 (𝜑 → (𝑇:ℕ⟶(ℝ ↑𝑚 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑𝑚 𝑋)))
8480, 83mpbird 249 . . 3 (𝜑𝑇:ℕ⟶(ℝ ↑𝑚 𝑋))
8572, 84jca 504 . 2 (𝜑 → (𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)))
8648, 13eleqtrd 2868 . . . . 5 (𝜑𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
87 fveq1 6498 . . . . . . . . . . . 12 (𝑙 = 𝐼 → (𝑙𝑗) = (𝐼𝑗))
8887coeq2d 5583 . . . . . . . . . . 11 (𝑙 = 𝐼 → ([,) ∘ (𝑙𝑗)) = ([,) ∘ (𝐼𝑗)))
8988fveq1d 6501 . . . . . . . . . 10 (𝑙 = 𝐼 → (([,) ∘ (𝑙𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
9089ixpeq2dv 8275 . . . . . . . . 9 (𝑙 = 𝐼X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9190adantr 473 . . . . . . . 8 ((𝑙 = 𝐼𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9291iuneq2dv 4815 . . . . . . 7 (𝑙 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9392sseq2d 3889 . . . . . 6 (𝑙 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9493elrab 3595 . . . . 5 (𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)} ↔ (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9586, 94sylib 210 . . . 4 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9695simprd 488 . . 3 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9756adantr 473 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
98 simpr 477 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
9997, 98fvovco 40885 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
100 mptexg 6810 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10163, 100syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
102101adantr 473 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10370, 102fvmpt2d 6607 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
104 fvexd 6514 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ V)
105103, 104fvmpt2d 6607 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) = (1st ‘((𝐼𝑗)‘𝑘)))
106105eqcomd 2784 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = ((𝐵𝑗)‘𝑘))
107 mptexg 6810 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
10863, 107syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
109108adantr 473 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
11082, 109fvmpt2d 6607 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
111 fvexd 6514 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ V)
112110, 111fvmpt2d 6607 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) = (2nd ‘((𝐼𝑗)‘𝑘)))
113112eqcomd 2784 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = ((𝑇𝑗)‘𝑘))
114106, 113oveq12d 6994 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11599, 114eqtrd 2814 . . . . 5 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
116115ixpeq2dva 8274 . . . 4 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
117116iuneq2dv 4815 . . 3 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11896, 117sseqtrd 3897 . 2 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
119 ovncvr2.l . . . . . . . 8 𝐿 = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)))
120119a1i 11 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → 𝐿 = ( ∈ ((ℝ × ℝ) ↑𝑚 𝑋) ↦ ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘))))
121 coeq2 5579 . . . . . . . . . . . . 13 ( = (𝐼𝑗) → ([,) ∘ ) = ([,) ∘ (𝐼𝑗)))
122121fveq1d 6501 . . . . . . . . . . . 12 ( = (𝐼𝑗) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
123122ad2antlr 714 . . . . . . . . . . 11 (((𝜑 = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
124123adantllr 706 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
12599adantlr 702 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
126114adantlr 702 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
127124, 125, 1263eqtrd 2818 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
128127fveq2d 6503 . . . . . . . 8 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (vol‘(([,) ∘ )‘𝑘)) = (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
129128prodeq2dv 15137 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) → ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
13063adantr 473 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
13169fvmpt2 6605 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
13253, 102, 131syl2anc 576 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
133132feq1d 6329 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝐵𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
13460, 133mpbird 249 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗):𝑋⟶ℝ)
135134adantr 473 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑗):𝑋⟶ℝ)
136135, 98ffvelrnd 6677 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) ∈ ℝ)
13781fvmpt2 6605 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
13853, 109, 137syl2anc 576 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
139138feq1d 6329 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝑇𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
14075, 139mpbird 249 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗):𝑋⟶ℝ)
141140adantr 473 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑇𝑗):𝑋⟶ℝ)
142141, 98ffvelrnd 6677 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) ∈ ℝ)
143 volicore 42300 . . . . . . . . 9 ((((𝐵𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
144136, 142, 143syl2anc 576 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
145130, 144fprodrecl 15167 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
146120, 129, 54, 145fvmptd 6601 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
147146eqcomd 2784 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) = (𝐿‘(𝐼𝑗)))
148147mpteq2dva 5022 . . . 4 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
149148fveq2d 6503 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
15047simprd 488 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
151149, 150eqbrtrd 4951 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15285, 118, 151jca31 507 1 (𝜑 → (((𝐵:ℕ⟶(ℝ ↑𝑚 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑𝑚 𝑋)) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  {crab 3092  Vcvv 3415  wss 3829  𝒫 cpw 4422   ciun 4792   class class class wbr 4929  cmpt 5008   × cxp 5405  ccom 5411  wf 6184  cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  𝑚 cmap 8206  Xcixp 8259  Fincfn 8306  cr 10334  cle 10475  cn 11439  +crp 12204   +𝑒 cxad 12322  [,)cico 12556  cprod 15119  volcvol 23767  Σ^csumge0 42081  voln*covoln 42255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279  ax-inf2 8898  ax-cnex 10391  ax-resscn 10392  ax-1cn 10393  ax-icn 10394  ax-addcl 10395  ax-addrcl 10396  ax-mulcl 10397  ax-mulrcl 10398  ax-mulcom 10399  ax-addass 10400  ax-mulass 10401  ax-distr 10402  ax-i2m1 10403  ax-1ne0 10404  ax-1rid 10405  ax-rnegex 10406  ax-rrecex 10407  ax-cnre 10408  ax-pre-lttri 10409  ax-pre-lttrn 10410  ax-pre-ltadd 10411  ax-pre-mulgt0 10412  ax-pre-sup 10413  ax-addf 10414  ax-mulf 10415
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-of 7227  df-om 7397  df-1st 7501  df-2nd 7502  df-tpos 7695  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-1o 7905  df-2o 7906  df-oadd 7909  df-er 8089  df-map 8208  df-pm 8209  df-ixp 8260  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-fi 8670  df-sup 8701  df-inf 8702  df-oi 8769  df-dju 9124  df-card 9162  df-pnf 10476  df-mnf 10477  df-xr 10478  df-ltxr 10479  df-le 10480  df-sub 10672  df-neg 10673  df-div 11099  df-nn 11440  df-2 11503  df-3 11504  df-4 11505  df-5 11506  df-6 11507  df-7 11508  df-8 11509  df-9 11510  df-n0 11708  df-z 11794  df-dec 11912  df-uz 12059  df-q 12163  df-rp 12205  df-xneg 12324  df-xadd 12325  df-xmul 12326  df-ioo 12558  df-ico 12560  df-icc 12561  df-fz 12709  df-fzo 12850  df-fl 12977  df-seq 13185  df-exp 13245  df-hash 13506  df-cj 14319  df-re 14320  df-im 14321  df-sqrt 14455  df-abs 14456  df-clim 14706  df-rlim 14707  df-sum 14904  df-prod 15120  df-struct 16341  df-ndx 16342  df-slot 16343  df-base 16345  df-sets 16346  df-ress 16347  df-plusg 16434  df-mulr 16435  df-starv 16436  df-tset 16440  df-ple 16441  df-ds 16443  df-unif 16444  df-rest 16552  df-0g 16571  df-topgen 16573  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-grp 17894  df-minusg 17895  df-subg 18060  df-cmn 18668  df-abl 18669  df-mgp 18963  df-ur 18975  df-ring 19022  df-cring 19023  df-oppr 19096  df-dvdsr 19114  df-unit 19115  df-invr 19145  df-dvr 19156  df-drng 19227  df-psmet 20239  df-xmet 20240  df-met 20241  df-bl 20242  df-mopn 20243  df-cnfld 20248  df-top 21206  df-topon 21223  df-bases 21258  df-cmp 21699  df-ovol 23768  df-vol 23769
This theorem is referenced by:  hspmbllem3  42347
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