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Theorem ovncvr2 46626
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 4009 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐴 → (𝑎 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)))
32rabbidv 3444 . . . . . . . . . . . . . . . 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 7466 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (ℝ ↑m 𝑋) ∈ V)
65, 4ssexd 5324 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ V)
7 elpwg 4603 . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . 16 (𝜑𝐴 ∈ 𝒫 (ℝ ↑m 𝑋))
10 ovex 7464 . . . . . . . . . . . . . . . . . 18 (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈ V
1110rabex 5339 . . . . . . . . . . . . . . . . 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 7039 . . . . . . . . . . . . . . 15 (𝜑 → (𝐶𝐴) = {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
14 ssrab2 4080 . . . . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (𝐶𝑎) = (𝐶𝐴))
2019eleq2d 2827 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = 𝐴 → (𝑖 ∈ (𝐶𝑎) ↔ 𝑖 ∈ (𝐶𝐴)))
21 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴))
2221oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))
2322breq2d 5155 . . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . . 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 3438 . . . . . . . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . . . . . . . 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 45357 . . . . . . . . . . . . . . . . . . . . 21 + ∈ V
2827mptex 7243 . . . . . . . . . . . . . . . . . . . 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 7039 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐷𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}))
31 oveq2 7439 . . . . . . . . . . . . . . . . . . . . 21 (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
3231breq2d 5155 . . . . . . . . . . . . . . . . . . . 20 (𝑟 = 𝐸 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
3332rabbidv 3444 . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . . . . . . . 20 (𝐶𝐴) ∈ V
3736rabex 5339 . . . . . . . . . . . . . . . . . . 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 7023 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝐷𝐴)‘𝐸) = {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
4017, 39eleqtrd 2843 . . . . . . . . . . . . . . . 16 (𝜑𝐼 ∈ {𝑖 ∈ (𝐶𝐴) ∣ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)})
41 fveq1 6905 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝐼 → (𝑖𝑗) = (𝐼𝑗))
4241fveq2d 6910 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝐿‘(𝑖𝑗)) = (𝐿‘(𝐼𝑗)))
4342mpteq2dv 5244 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
4443fveq2d 6910 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
4544breq1d 5153 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝐼 → ((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4645elrab 3692 . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . 15 (𝜑 → (𝐼 ∈ (𝐶𝐴) ∧ (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)))
4847simpld 494 . . . . . . . . . . . . . 14 (𝜑𝐼 ∈ (𝐶𝐴))
4916, 48sseldd 3984 . . . . . . . . . . . . 13 (𝜑𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ))
50 elmapi 8889 . . . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ) ↑m 𝑋))
53 simpr 484 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → 𝑗 ∈ ℕ)
5452, 53ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋))
55 elmapi 8889 . . . . . . . . . 10 ((𝐼𝑗) ∈ ((ℝ × ℝ) ↑m 𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5654, 55syl 17 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
5756ffvelcdmda 7104 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ))
58 xp1st 8046 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
5957, 58syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
6059fmpttd 7135 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
61 reex 11246 . . . . . . . . 9 ℝ ∈ V
6261a1i 11 . . . . . . . 8 (𝜑 → ℝ ∈ V)
63 ovncvr2.x . . . . . . . 8 (𝜑𝑋 ∈ Fin)
64 elmapg 8879 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6562, 63, 64syl2anc 584 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6665adantr 480 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
6760, 66mpbird 257 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
6867fmpttd 7135 . . . 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 6720 . . . 4 (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
7268, 71mpbird 257 . . 3 (𝜑𝐵:ℕ⟶(ℝ ↑m 𝑋))
73 xp2nd 8047 . . . . . . . 8 (((𝐼𝑗)‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7457, 73syl 17 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ ℝ)
7574fmpttd 7135 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ)
76 elmapg 8879 . . . . . . . 8 ((ℝ ∈ V ∧ 𝑋 ∈ Fin) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7762, 63, 76syl2anc 584 . . . . . . 7 (𝜑 → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7877adantr 480 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → ((𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
7975, 78mpbird 257 . . . . 5 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋))
8079fmpttd 7135 . . . 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 6720 . . . 4 (𝜑 → (𝑇:ℕ⟶(ℝ ↑m 𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘)))):ℕ⟶(ℝ ↑m 𝑋)))
8480, 83mpbird 257 . . 3 (𝜑𝑇:ℕ⟶(ℝ ↑m 𝑋))
8572, 84jca 511 . 2 (𝜑 → (𝐵:ℕ⟶(ℝ ↑m 𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m 𝑋)))
8648, 13eleqtrd 2843 . . . . 5 (𝜑𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘)})
87 fveq1 6905 . . . . . . . . . . . 12 (𝑙 = 𝐼 → (𝑙𝑗) = (𝐼𝑗))
8887coeq2d 5873 . . . . . . . . . . 11 (𝑙 = 𝐼 → ([,) ∘ (𝑙𝑗)) = ([,) ∘ (𝐼𝑗)))
8988fveq1d 6908 . . . . . . . . . 10 (𝑙 = 𝐼 → (([,) ∘ (𝑙𝑗))‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
9089ixpeq2dv 8953 . . . . . . . . 9 (𝑙 = 𝐼X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9190adantr 480 . . . . . . . 8 ((𝑙 = 𝐼𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9291iuneq2dv 5016 . . . . . . 7 (𝑙 = 𝐼 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9392sseq2d 4016 . . . . . 6 (𝑙 = 𝐼 → (𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝑙𝑗))‘𝑘) ↔ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9493elrab 3692 . . . . 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 218 . . . 4 (𝜑 → (𝐼 ∈ (((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∧ 𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘)))
9695simprd 495 . . 3 (𝜑𝐴 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘))
9756adantr 480 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐼𝑗):𝑋⟶(ℝ × ℝ))
98 simpr 484 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → 𝑘𝑋)
9997, 98fvovco 45198 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
100 mptexg 7241 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10163, 100syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
102101adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V)
10370, 102fvmpt2d 7029 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
104 fvexd 6921 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) ∈ V)
105103, 104fvmpt2d 7029 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) = (1st ‘((𝐼𝑗)‘𝑘)))
106105eqcomd 2743 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (1st ‘((𝐼𝑗)‘𝑘)) = ((𝐵𝑗)‘𝑘))
107 mptexg 7241 . . . . . . . . . . . 12 (𝑋 ∈ Fin → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
10863, 107syl 17 . . . . . . . . . . 11 (𝜑 → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
109108adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ ℕ) → (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V)
11082, 109fvmpt2d 7029 . . . . . . . . 9 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
111 fvexd 6921 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) ∈ V)
112110, 111fvmpt2d 7029 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) = (2nd ‘((𝐼𝑗)‘𝑘)))
113112eqcomd 2743 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (2nd ‘((𝐼𝑗)‘𝑘)) = ((𝑇𝑗)‘𝑘))
114106, 113oveq12d 7449 . . . . . 6 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11599, 114eqtrd 2777 . . . . 5 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
116115ixpeq2dva 8952 . . . 4 ((𝜑𝑗 ∈ ℕ) → X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
117116iuneq2dv 5016 . . 3 (𝜑 𝑗 ∈ ℕ X𝑘𝑋 (([,) ∘ (𝐼𝑗))‘𝑘) = 𝑗 ∈ ℕ X𝑘𝑋 (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
11896, 117sseqtrd 4020 . 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 5869 . . . . . . . . . . . . 13 ( = (𝐼𝑗) → ([,) ∘ ) = ([,) ∘ (𝐼𝑗)))
122121fveq1d 6908 . . . . . . . . . . . 12 ( = (𝐼𝑗) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
123122ad2antlr 727 . . . . . . . . . . 11 (((𝜑 = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
124123adantllr 719 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (([,) ∘ (𝐼𝑗))‘𝑘))
12599adantlr 715 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ (𝐼𝑗))‘𝑘) = ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))))
126114adantlr 715 . . . . . . . . . 10 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → ((1st ‘((𝐼𝑗)‘𝑘))[,)(2nd ‘((𝐼𝑗)‘𝑘))) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
127124, 125, 1263eqtrd 2781 . . . . . . . . 9 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (([,) ∘ )‘𝑘) = (((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))
128127fveq2d 6910 . . . . . . . 8 ((((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) ∧ 𝑘𝑋) → (vol‘(([,) ∘ )‘𝑘)) = (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
129128prodeq2dv 15958 . . . . . . 7 (((𝜑𝑗 ∈ ℕ) ∧ = (𝐼𝑗)) → ∏𝑘𝑋 (vol‘(([,) ∘ )‘𝑘)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
13063adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ) → 𝑋 ∈ Fin)
13169fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
13253, 102, 131syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗) = (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))))
133132feq1d 6720 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝐵𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (1st ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
13460, 133mpbird 257 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝐵𝑗):𝑋⟶ℝ)
135134adantr 480 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝐵𝑗):𝑋⟶ℝ)
136135, 98ffvelcdmd 7105 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝐵𝑗)‘𝑘) ∈ ℝ)
13781fvmpt2 7027 . . . . . . . . . . . . . 14 ((𝑗 ∈ ℕ ∧ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))) ∈ V) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
13853, 109, 137syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗) = (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))))
139138feq1d 6720 . . . . . . . . . . . 12 ((𝜑𝑗 ∈ ℕ) → ((𝑇𝑗):𝑋⟶ℝ ↔ (𝑘𝑋 ↦ (2nd ‘((𝐼𝑗)‘𝑘))):𝑋⟶ℝ))
14075, 139mpbird 257 . . . . . . . . . . 11 ((𝜑𝑗 ∈ ℕ) → (𝑇𝑗):𝑋⟶ℝ)
141140adantr 480 . . . . . . . . . 10 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (𝑇𝑗):𝑋⟶ℝ)
142141, 98ffvelcdmd 7105 . . . . . . . . 9 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → ((𝑇𝑗)‘𝑘) ∈ ℝ)
143 volicore 46596 . . . . . . . . 9 ((((𝐵𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
144136, 142, 143syl2anc 584 . . . . . . . 8 (((𝜑𝑗 ∈ ℕ) ∧ 𝑘𝑋) → (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
145130, 144fprodrecl 15989 . . . . . . 7 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) ∈ ℝ)
146120, 129, 54, 145fvmptd 7023 . . . . . 6 ((𝜑𝑗 ∈ ℕ) → (𝐿‘(𝐼𝑗)) = ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))
147146eqcomd 2743 . . . . 5 ((𝜑𝑗 ∈ ℕ) → ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))) = (𝐿‘(𝐼𝑗)))
148147mpteq2dva 5242 . . . 4 (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗))))
149148fveq2d 6910 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) = (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))))
15047simprd 495 . . 3 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
151149, 150eqbrtrd 5165 . 2 (𝜑 → (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘𝑋 (vol‘(((𝐵𝑗)‘𝑘)[,)((𝑇𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))
15285, 118, 151jca31 514 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 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  wss 3951  𝒫 cpw 4600   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  m cmap 8866  Xcixp 8937  Fincfn 8985  cr 11154  cle 11296  cn 12266  +crp 13034   +𝑒 cxad 13152  [,)cico 13389  cprod 15939  volcvol 25498  Σ^csumge0 46377  voln*covoln 46551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-prod 15940  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500
This theorem is referenced by:  hspmbllem3  46643
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