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Theorem ovncvr2 46028
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 4007 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐴 → (𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
32rabbidv 3438 . . . . . . . . . . . . . . . 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 7461 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
65, 4ssexd 5328 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
7 elpwg 4609 . . . . . . . . . . . . . . . . . 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 256 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
10 ovex 7459 . . . . . . . . . . . . . . . . . 18 (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
1110rabex 5338 . . . . . . . . . . . . . . . . 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 7033 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
14 ssrab2 4077 . . . . . . . . . . . . . . . 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 4020 . . . . . . . . . . . . . 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 6902 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (𝐶𝑎) = (𝐶𝐴))
2019eleq2d 2815 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝐴 → (𝑖 ∈ (𝐶𝑎) ↔ 𝑖 ∈ (𝐶𝐴)))
21 fveq2 6902 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴))
2221oveq1d 7441 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))
2322breq2d 5164 . . . . . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . . . . . 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 3432 . . . . . . . . . . . . . . . . . . . 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 5254 . . . . . . . . . . . . . . . . . . 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 44757 . . . . . . . . . . . . . . . . . . . . 21 + ∈ V
2827mptex 7241 . . . . . . . . . . . . . . . . . . . 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 7033 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
31 oveq2 7434 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
3231breq2d 5164 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
3332rabbidv 3438 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 6915 . . . . . . . . . . . . . . . . . . . 20 (𝐶𝐴) ∈ V
3736rabex 5338 . . . . . . . . . . . . . . . . . . 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 7017 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐷𝐴)‘𝐸) = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
4017, 39eleqtrd 2831 . . . . . . . . . . . . . . . 16 (𝜑𝐼 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
41 fveq1 6901 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4241fveq2d 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝐿‘(𝑖𝑗)) = (𝐿‘(𝐼𝑗)))
4342mpteq2dv 5254 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
4443fveq2d 6906 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
4544breq1d 5162 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4645elrab 3684 . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . 15 (𝜑 → (𝐼 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4847simpld 493 . . . . . . . . . . . . . 14 (𝜑𝐼 ∈ (𝐶𝐴))
4916, 48sseldd 3983 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
50 elmapi 8874 . . . . . . . . . . . . 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 479 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
53 simpr 483 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
5452, 53ffvelcdmd 7100 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
55 elmapi 8874 . . . . . . . . . 10 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5654, 55syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5756ffvelcdmda 7099 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ))
58 xp1st 8031 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
5957, 58syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
6059fmpttd 7130 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
61 reex 11237 . . . . . . . . 9 ℝ ∈ V
6261a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
63 ovncvr2.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
64 elmapg 8864 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6562, 63, 64syl2anc 582 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6665adantr 479 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6760, 66mpbird 256 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
6867fmpttd 7130 . . . 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 6712 . . . 4 (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
7268, 71mpbird 256 . . 3 (𝜑𝐵:ℕ⟶(ℝ ↑m 𝑋))
73 xp2nd 8032 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7457, 73syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7574fmpttd 7130 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
76 elmapg 8864 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7762, 63, 76syl2anc 582 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7877adantr 479 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7975, 78mpbird 256 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
8079fmpttd 7130 . . . 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 6712 . . . 4 (𝜑 → (𝑇:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
8480, 83mpbird 256 . . 3 (𝜑𝑇:ℕ⟶(ℝ ↑m 𝑋))
8572, 84jca 510 . 2 (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m 𝑋)))
8648, 13eleqtrd 2831 . . . . 5 (𝜑𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
87 fveq1 6901 . . . . . . . . . . . 12 (𝑙 = 𝐼 → (𝑙𝑗) = (𝐼𝑗))
8887coeq2d 5869 . . . . . . . . . . 11 (𝑙 = 𝐼 → ([,) ∘ (𝑙𝑗)) = ([,) ∘ (𝐼𝑗)))
8988fveq1d 6904 . . . . . . . . . 10 (𝑙 = 𝐼 → (([,) ∘ (𝑙𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
9089ixpeq2dv 8938 . . . . . . . . 9 (𝑙 = 𝐼X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9190adantr 479 . . . . . . . 8 ((𝑙 = 𝐼𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9291iuneq2dv 5024 . . . . . . 7 (𝑙 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9392sseq2d 4014 . . . . . 6 (𝑙 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9493elrab 3684 . . . . 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 217 . . . 4 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9695simprd 494 . . 3 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9756adantr 479 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
98 simpr 483 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
9997, 98fvovco 44596 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
100 mptexg 7239 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10163, 100syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
102101adantr 479 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10370, 102fvmpt2d 7023 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
104 fvexd 6917 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ V)
105103, 104fvmpt2d 7023 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) = (1st ‘((𝐼𝑗)‘𝑘)))
106105eqcomd 2734 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = ((𝐵𝑗)‘𝑘))
107 mptexg 7239 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
10863, 107syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
109108adantr 479 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
11082, 109fvmpt2d 7023 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
111 fvexd 6917 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ V)
112110, 111fvmpt2d 7023 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) = (2nd ‘((𝐼𝑗)‘𝑘)))
113112eqcomd 2734 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = ((𝑇𝑗)‘𝑘))
114106, 113oveq12d 7444 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11599, 114eqtrd 2768 . . . . 5 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
116115ixpeq2dva 8937 . . . 4 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
117116iuneq2dv 5024 . . 3 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11896, 117sseqtrd 4022 . 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 5865 . . . . . . . . . . . . 13 ( = (𝐼𝑗) → ([,) ∘ ) = ([,) ∘ (𝐼𝑗)))
122121fveq1d 6904 . . . . . . . . . . . 12 ( = (𝐼𝑗) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
123122ad2antlr 725 . . . . . . . . . . 11 (((𝜑 = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
124123adantllr 717 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
12599adantlr 713 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
126114adantlr 713 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
127124, 125, 1263eqtrd 2772 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
128127fveq2d 6906 . . . . . . . 8 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (vol‘(([,) ∘ )‘𝑘)) = (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
129128prodeq2dv 15907 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) → ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
13063adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
13169fvmpt2 7021 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
13253, 102, 131syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
133132feq1d 6712 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝐵𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
13460, 133mpbird 256 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗):𝑋⟶ℝ)
135134adantr 479 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑗):𝑋⟶ℝ)
136135, 98ffvelcdmd 7100 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) ∈ ℝ)
13781fvmpt2 7021 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
13853, 109, 137syl2anc 582 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
139138feq1d 6712 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝑇𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
14075, 139mpbird 256 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗):𝑋⟶ℝ)
141140adantr 479 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑇𝑗):𝑋⟶ℝ)
142141, 98ffvelcdmd 7100 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) ∈ ℝ)
143 volicore 45998 . . . . . . . . 9 ((((𝐵𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
144136, 142, 143syl2anc 582 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
145130, 144fprodrecl 15937 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
146120, 129, 54, 145fvmptd 7017 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
147146eqcomd 2734 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) = (𝐿‘(𝐼𝑗)))
148147mpteq2dva 5252 . . . 4 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
149148fveq2d 6906 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
15047simprd 494 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
151149, 150eqbrtrd 5174 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15285, 118, 151jca31 513 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 205  wa 394   = wceq 1533  wcel 2098  {crab 3430  Vcvv 3473  wss 3949  𝒫 cpw 4606   ciun 5000   class class class wbr 5152  cmpt 5235   × cxp 5680  ccom 5686  wf 6549  cfv 6553  (class class class)co 7426  1st c1st 7997  2nd c2nd 7998  m cmap 8851  Xcixp 8922  Fincfn 8970  cr 11145  cle 11287  cn 12250  +crp 13014   +𝑒 cxad 13130  [,)cico 13366  cprod 15889  volcvol 25412  Σ^csumge0 45779  voln*covoln 45953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224  ax-addf 11225  ax-mulf 11226
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-of 7691  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-map 8853  df-pm 8854  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-fi 9442  df-sup 9473  df-inf 9474  df-oi 9541  df-dju 9932  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-q 12971  df-rp 13015  df-xneg 13132  df-xadd 13133  df-xmul 13134  df-ioo 13368  df-ico 13370  df-icc 13371  df-fz 13525  df-fzo 13668  df-fl 13797  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-rlim 15473  df-sum 15673  df-prod 15890  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-starv 17255  df-tset 17259  df-ple 17260  df-ds 17262  df-unif 17263  df-rest 17411  df-0g 17430  df-topgen 17432  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-minusg 18901  df-subg 19085  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-cring 20183  df-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-dvr 20347  df-drng 20633  df-psmet 21278  df-xmet 21279  df-met 21280  df-bl 21281  df-mopn 21282  df-cnfld 21287  df-top 22816  df-topon 22833  df-bases 22869  df-cmp 23311  df-ovol 25413  df-vol 25414
This theorem is referenced by:  hspmbllem3  46045
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