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Theorem ovncvr2 43176
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 3978 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐴 → (𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
32rabbidv 3465 . . . . . . . . . . . . . . . 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 7184 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
65, 4ssexd 5214 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
7 elpwg 4525 . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
10 ovex 7182 . . . . . . . . . . . . . . . . . 18 (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
1110rabex 5221 . . . . . . . . . . . . . . . . 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 6782 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
14 ssrab2 4042 . . . . . . . . . . . . . . . 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 3991 . . . . . . . . . . . . . 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 6661 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (𝐶𝑎) = (𝐶𝐴))
2019eleq2d 2901 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝐴 → (𝑖 ∈ (𝐶𝑎) ↔ 𝑖 ∈ (𝐶𝐴)))
21 fveq2 6661 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴))
2221oveq1d 7164 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))
2322breq2d 5064 . . . . . . . . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . . . . . . . . . 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 3461 . . . . . . . . . . . . . . . . . . . 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 5148 . . . . . . . . . . . . . . . . . . 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 41904 . . . . . . . . . . . . . . . . . . . . 21 + ∈ V
2827mptex 6977 . . . . . . . . . . . . . . . . . . . 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 6782 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
31 oveq2 7157 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
3231breq2d 5064 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
3332rabbidv 3465 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 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 6674 . . . . . . . . . . . . . . . . . . . 20 (𝐶𝐴) ∈ V
3736rabex 5221 . . . . . . . . . . . . . . . . . . 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 6766 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐷𝐴)‘𝐸) = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
4017, 39eleqtrd 2918 . . . . . . . . . . . . . . . 16 (𝜑𝐼 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
41 fveq1 6660 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4241fveq2d 6665 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝐿‘(𝑖𝑗)) = (𝐿‘(𝐼𝑗)))
4342mpteq2dv 5148 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
4443fveq2d 6665 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
4544breq1d 5062 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4645elrab 3666 . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . 15 (𝜑 → (𝐼 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4847simpld 498 . . . . . . . . . . . . . 14 (𝜑𝐼 ∈ (𝐶𝐴))
4916, 48sseldd 3954 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
50 elmapi 8424 . . . . . . . . . . . . 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 484 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
53 simpr 488 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
5452, 53ffvelrnd 6843 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
55 elmapi 8424 . . . . . . . . . 10 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5654, 55syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5756ffvelrnda 6842 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ))
58 xp1st 7716 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
5957, 58syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
6059fmpttd 6870 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
61 reex 10626 . . . . . . . . 9 ℝ ∈ V
6261a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
63 ovncvr2.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
64 elmapg 8415 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6562, 63, 64syl2anc 587 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6665adantr 484 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6760, 66mpbird 260 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
6867fmpttd 6870 . . . 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 6488 . . . 4 (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
7268, 71mpbird 260 . . 3 (𝜑𝐵:ℕ⟶(ℝ ↑m 𝑋))
73 xp2nd 7717 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7457, 73syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7574fmpttd 6870 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
76 elmapg 8415 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7762, 63, 76syl2anc 587 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7877adantr 484 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7975, 78mpbird 260 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
8079fmpttd 6870 . . . 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 6488 . . . 4 (𝜑 → (𝑇:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
8480, 83mpbird 260 . . 3 (𝜑𝑇:ℕ⟶(ℝ ↑m 𝑋))
8572, 84jca 515 . 2 (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m 𝑋)))
8648, 13eleqtrd 2918 . . . . 5 (𝜑𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
87 fveq1 6660 . . . . . . . . . . . 12 (𝑙 = 𝐼 → (𝑙𝑗) = (𝐼𝑗))
8887coeq2d 5720 . . . . . . . . . . 11 (𝑙 = 𝐼 → ([,) ∘ (𝑙𝑗)) = ([,) ∘ (𝐼𝑗)))
8988fveq1d 6663 . . . . . . . . . 10 (𝑙 = 𝐼 → (([,) ∘ (𝑙𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
9089ixpeq2dv 8473 . . . . . . . . 9 (𝑙 = 𝐼X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9190adantr 484 . . . . . . . 8 ((𝑙 = 𝐼𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9291iuneq2dv 4929 . . . . . . 7 (𝑙 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9392sseq2d 3985 . . . . . 6 (𝑙 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9493elrab 3666 . . . . 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 221 . . . 4 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9695simprd 499 . . 3 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9756adantr 484 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
98 simpr 488 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
9997, 98fvovco 41746 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
100 mptexg 6975 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10163, 100syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
102101adantr 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10370, 102fvmpt2d 6772 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
104 fvexd 6676 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ V)
105103, 104fvmpt2d 6772 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) = (1st ‘((𝐼𝑗)‘𝑘)))
106105eqcomd 2830 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = ((𝐵𝑗)‘𝑘))
107 mptexg 6975 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
10863, 107syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
109108adantr 484 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
11082, 109fvmpt2d 6772 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
111 fvexd 6676 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ V)
112110, 111fvmpt2d 6772 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) = (2nd ‘((𝐼𝑗)‘𝑘)))
113112eqcomd 2830 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = ((𝑇𝑗)‘𝑘))
114106, 113oveq12d 7167 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11599, 114eqtrd 2859 . . . . 5 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
116115ixpeq2dva 8472 . . . 4 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
117116iuneq2dv 4929 . . 3 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11896, 117sseqtrd 3993 . 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 5716 . . . . . . . . . . . . 13 ( = (𝐼𝑗) → ([,) ∘ ) = ([,) ∘ (𝐼𝑗)))
122121fveq1d 6663 . . . . . . . . . . . 12 ( = (𝐼𝑗) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
123122ad2antlr 726 . . . . . . . . . . 11 (((𝜑 = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
124123adantllr 718 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
12599adantlr 714 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
126114adantlr 714 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
127124, 125, 1263eqtrd 2863 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
128127fveq2d 6665 . . . . . . . 8 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (vol‘(([,) ∘ )‘𝑘)) = (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
129128prodeq2dv 15277 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) → ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
13063adantr 484 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
13169fvmpt2 6770 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
13253, 102, 131syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
133132feq1d 6488 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝐵𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
13460, 133mpbird 260 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗):𝑋⟶ℝ)
135134adantr 484 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑗):𝑋⟶ℝ)
136135, 98ffvelrnd 6843 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) ∈ ℝ)
13781fvmpt2 6770 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
13853, 109, 137syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
139138feq1d 6488 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝑇𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
14075, 139mpbird 260 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗):𝑋⟶ℝ)
141140adantr 484 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑇𝑗):𝑋⟶ℝ)
142141, 98ffvelrnd 6843 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) ∈ ℝ)
143 volicore 43146 . . . . . . . . 9 ((((𝐵𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
144136, 142, 143syl2anc 587 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
145130, 144fprodrecl 15307 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
146120, 129, 54, 145fvmptd 6766 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
147146eqcomd 2830 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) = (𝐿‘(𝐼𝑗)))
148147mpteq2dva 5147 . . . 4 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
149148fveq2d 6665 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
15047simprd 499 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
151149, 150eqbrtrd 5074 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15285, 118, 151jca31 518 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 209  wa 399   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  wss 3919  𝒫 cpw 4522   ciun 4905   class class class wbr 5052  cmpt 5132   × cxp 5540  ccom 5546  wf 6339  cfv 6343  (class class class)co 7149  1st c1st 7682  2nd c2nd 7683  m cmap 8402  Xcixp 8457  Fincfn 8505  cr 10534  cle 10674  cn 11634  +crp 12386   +𝑒 cxad 12502  [,)cico 12737  cprod 15259  volcvol 24070  Σ^csumge0 42927  voln*covoln 43101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613  ax-addf 10614  ax-mulf 10615
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-tpos 7888  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-ixp 8458  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-fi 8872  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-q 12346  df-rp 12387  df-xneg 12504  df-xadd 12505  df-xmul 12506  df-ioo 12739  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-rlim 14846  df-sum 15043  df-prod 15260  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-rest 16696  df-0g 16715  df-topgen 16717  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107  df-subg 18276  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-ring 19299  df-cring 19300  df-oppr 19376  df-dvdsr 19394  df-unit 19395  df-invr 19425  df-dvr 19436  df-drng 19504  df-psmet 20537  df-xmet 20538  df-met 20539  df-bl 20540  df-mopn 20541  df-cnfld 20546  df-top 21502  df-topon 21519  df-bases 21554  df-cmp 21995  df-ovol 24071  df-vol 24072
This theorem is referenced by:  hspmbllem3  43193
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