Proof of Theorem ovncvr2
| Step | Hyp | Ref
| Expression |
| 1 | | ovncvr2.c |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = (𝑎 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝑎 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 2 | | sseq1 4009 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝐴 → (𝑎 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘))) |
| 3 | 2 | rabbidv 3444 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝐴 → {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝑎 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} = {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 4 | | ovncvr2.a |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐴 ⊆ (ℝ ↑m 𝑋)) |
| 5 | | ovexd 7466 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (ℝ
↑m 𝑋)
∈ V) |
| 6 | 5, 4 | ssexd 5324 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐴 ∈ V) |
| 7 | | elpwg 4603 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 (ℝ
↑m 𝑋)
↔ 𝐴 ⊆ (ℝ
↑m 𝑋))) |
| 8 | 6, 7 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐴 ∈ 𝒫 (ℝ
↑m 𝑋)
↔ 𝐴 ⊆ (ℝ
↑m 𝑋))) |
| 9 | 4, 8 | mpbird 257 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐴 ∈ 𝒫 (ℝ
↑m 𝑋)) |
| 10 | | ovex 7464 |
. . . . . . . . . . . . . . . . . 18
⊢
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ) ∈
V |
| 11 | 10 | rabex 5339 |
. . . . . . . . . . . . . . . . 17
⊢ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V |
| 12 | 11 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ∈ V) |
| 13 | 1, 3, 9, 12 | fvmptd3 7039 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶‘𝐴) = {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 14 | | ssrab2 4080 |
. . . . . . . . . . . . . . . 16
⊢ {𝑙 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ∣ 𝐴 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) |
| 15 | 14 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ⊆ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)) |
| 16 | 13, 15 | eqsstrd 4018 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐶‘𝐴) ⊆ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)) |
| 17 | | ovncvr2.i |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐼 ∈ ((𝐷‘𝐴)‘𝐸)) |
| 18 | | ovncvr2.d |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐷 = (𝑎 ∈ 𝒫 (ℝ ↑m
𝑋) ↦ (𝑟 ∈ ℝ+
↦ {𝑖 ∈ (𝐶‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)})) |
| 19 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑎 = 𝐴 → (𝐶‘𝑎) = (𝐶‘𝐴)) |
| 20 | 19 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = 𝐴 → (𝑖 ∈ (𝐶‘𝑎) ↔ 𝑖 ∈ (𝐶‘𝐴))) |
| 21 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑎 = 𝐴 → ((voln*‘𝑋)‘𝑎) = ((voln*‘𝑋)‘𝐴)) |
| 22 | 21 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑎 = 𝐴 → (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)) |
| 23 | 22 | breq2d 5155 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = 𝐴 →
((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟) ↔
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟))) |
| 24 | 20, 23 | anbi12d 632 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 = 𝐴 → ((𝑖 ∈ (𝐶‘𝑎) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)) ↔ (𝑖 ∈ (𝐶‘𝐴) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)))) |
| 25 | 24 | rabbidva2 3438 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = 𝐴 → {𝑖 ∈ (𝐶‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)} = {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) |
| 26 | 25 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝐴 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝑎) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝑎) +𝑒 𝑟)}) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)})) |
| 27 | | rpex 45357 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
ℝ+ ∈ V |
| 28 | 27 | mptex 7243 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 ∈ ℝ+
↦ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V |
| 29 | 28 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)}) ∈ V) |
| 30 | 18, 26, 9, 29 | fvmptd3 7039 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐷‘𝐴) = (𝑟 ∈ ℝ+ ↦ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)})) |
| 31 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 = 𝐸 → (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) = (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
| 32 | 31 | breq2d 5155 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 = 𝐸 →
((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟) ↔
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) |
| 33 | 32 | rabbidv 3444 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = 𝐸 → {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)} = {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)}) |
| 34 | 33 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑟 = 𝐸) → {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝑟)} = {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)}) |
| 35 | | ovncvr2.e |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
| 36 | | fvex 6919 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐶‘𝐴) ∈ V |
| 37 | 36 | rabex 5339 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V |
| 38 | 37 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ∈ V) |
| 39 | 30, 34, 35, 38 | fvmptd 7023 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐷‘𝐴)‘𝐸) = {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)}) |
| 40 | 17, 39 | eleqtrd 2843 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐼 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)}) |
| 41 | | fveq1 6905 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗)) |
| 42 | 41 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝐼 → (𝐿‘(𝑖‘𝑗)) = (𝐿‘(𝐼‘𝑗))) |
| 43 | 42 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) |
| 44 | 43 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝐼 →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))) |
| 45 | 44 | breq1d 5153 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝐼 →
((Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸) ↔
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) |
| 46 | 45 | elrab 3692 |
. . . . . . . . . . . . . . . 16
⊢ (𝐼 ∈ {𝑖 ∈ (𝐶‘𝐴) ∣
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝑖‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)} ↔ (𝐼 ∈ (𝐶‘𝐴) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) |
| 47 | 40, 46 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐼 ∈ (𝐶‘𝐴) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) |
| 48 | 47 | simpld 494 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ (𝐶‘𝐴)) |
| 49 | 16, 48 | sseldd 3984 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)) |
| 50 | | elmapi 8889 |
. . . . . . . . . . . . 13
⊢ (𝐼 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) → 𝐼:ℕ⟶((ℝ
× ℝ) ↑m 𝑋)) |
| 51 | 49, 50 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ)
↑m 𝑋)) |
| 52 | 51 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐼:ℕ⟶((ℝ × ℝ)
↑m 𝑋)) |
| 53 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
| 54 | 52, 53 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑m 𝑋)) |
| 55 | | elmapi 8889 |
. . . . . . . . . 10
⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑m 𝑋)
→ (𝐼‘𝑗):𝑋⟶(ℝ ×
ℝ)) |
| 56 | 54, 55 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ ×
ℝ)) |
| 57 | 56 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) ∈ (ℝ ×
ℝ)) |
| 58 | | xp1st 8046 |
. . . . . . . 8
⊢ (((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ) →
(1st ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ) |
| 59 | 57, 58 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ) |
| 60 | 59 | fmpttd 7135 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ) |
| 61 | | reex 11246 |
. . . . . . . . 9
⊢ ℝ
∈ V |
| 62 | 61 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ ∈
V) |
| 63 | | ovncvr2.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ Fin) |
| 64 | | elmapg 8879 |
. . . . . . . 8
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
Fin) → ((𝑘 ∈
𝑋 ↦ (1st
‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 65 | 62, 63, 64 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 66 | 65 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 67 | 60, 66 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋)) |
| 68 | 67 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ
↑m 𝑋)) |
| 69 | | ovncvr2.b |
. . . . . 6
⊢ 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))) |
| 70 | 69 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐵 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))))) |
| 71 | 70 | feq1d 6720 |
. . . 4
⊢ (𝜑 → (𝐵:ℕ⟶(ℝ ↑m
𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ
↑m 𝑋))) |
| 72 | 68, 71 | mpbird 257 |
. . 3
⊢ (𝜑 → 𝐵:ℕ⟶(ℝ ↑m
𝑋)) |
| 73 | | xp2nd 8047 |
. . . . . . . 8
⊢ (((𝐼‘𝑗)‘𝑘) ∈ (ℝ × ℝ) →
(2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ) |
| 74 | 57, 73 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ ℝ) |
| 75 | 74 | fmpttd 7135 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ) |
| 76 | | elmapg 8879 |
. . . . . . . 8
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
Fin) → ((𝑘 ∈
𝑋 ↦ (2nd
‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 77 | 62, 63, 76 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 78 | 77 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋) ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 79 | 75, 78 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ (ℝ ↑m 𝑋)) |
| 80 | 79 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ
↑m 𝑋)) |
| 81 | | ovncvr2.t |
. . . . . 6
⊢ 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))) |
| 82 | 81 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝑇 = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))))) |
| 83 | 82 | feq1d 6720 |
. . . 4
⊢ (𝜑 → (𝑇:ℕ⟶(ℝ ↑m
𝑋) ↔ (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))):ℕ⟶(ℝ
↑m 𝑋))) |
| 84 | 80, 83 | mpbird 257 |
. . 3
⊢ (𝜑 → 𝑇:ℕ⟶(ℝ ↑m
𝑋)) |
| 85 | 72, 84 | jca 511 |
. 2
⊢ (𝜑 → (𝐵:ℕ⟶(ℝ ↑m
𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m
𝑋))) |
| 86 | 48, 13 | eleqtrd 2843 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)}) |
| 87 | | fveq1 6905 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝐼 → (𝑙‘𝑗) = (𝐼‘𝑗)) |
| 88 | 87 | coeq2d 5873 |
. . . . . . . . . . 11
⊢ (𝑙 = 𝐼 → ([,) ∘ (𝑙‘𝑗)) = ([,) ∘ (𝐼‘𝑗))) |
| 89 | 88 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝑙 = 𝐼 → (([,) ∘ (𝑙‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 90 | 89 | ixpeq2dv 8953 |
. . . . . . . . 9
⊢ (𝑙 = 𝐼 → X𝑘 ∈ 𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 91 | 90 | adantr 480 |
. . . . . . . 8
⊢ ((𝑙 = 𝐼 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 92 | 91 | iuneq2dv 5016 |
. . . . . . 7
⊢ (𝑙 = 𝐼 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 93 | 92 | sseq2d 4016 |
. . . . . 6
⊢ (𝑙 = 𝐼 → (𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘) ↔ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))) |
| 94 | 93 | elrab 3692 |
. . . . 5
⊢ (𝐼 ∈ {𝑙 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∣ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑙‘𝑗))‘𝑘)} ↔ (𝐼 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∧ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))) |
| 95 | 86, 94 | sylib 218 |
. . . 4
⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ∧ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘))) |
| 96 | 95 | simprd 495 |
. . 3
⊢ (𝜑 → 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 97 | 56 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ ×
ℝ)) |
| 98 | | simpr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
| 99 | 97, 98 | fvovco 45198 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘)))) |
| 100 | | mptexg 7241 |
. . . . . . . . . . . 12
⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V) |
| 101 | 63, 100 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V) |
| 102 | 101 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V) |
| 103 | 70, 102 | fvmpt2d 7029 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))) |
| 104 | | fvexd 6921 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) ∈ V) |
| 105 | 103, 104 | fvmpt2d 7029 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑗)‘𝑘) = (1st ‘((𝐼‘𝑗)‘𝑘))) |
| 106 | 105 | eqcomd 2743 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = ((𝐵‘𝑗)‘𝑘)) |
| 107 | | mptexg 7241 |
. . . . . . . . . . . 12
⊢ (𝑋 ∈ Fin → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V) |
| 108 | 63, 107 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V) |
| 109 | 108 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V) |
| 110 | 82, 109 | fvmpt2d 7029 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))) |
| 111 | | fvexd 6921 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) ∈ V) |
| 112 | 110, 111 | fvmpt2d 7029 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑗)‘𝑘) = (2nd ‘((𝐼‘𝑗)‘𝑘))) |
| 113 | 112 | eqcomd 2743 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = ((𝑇‘𝑗)‘𝑘)) |
| 114 | 106, 113 | oveq12d 7449 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 115 | 99, 114 | eqtrd 2777 |
. . . . 5
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 116 | 115 | ixpeq2dva 8952 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 117 | 116 | iuneq2dv 5016 |
. . 3
⊢ (𝜑 → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 118 | 96, 117 | sseqtrd 4020 |
. 2
⊢ (𝜑 → 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 119 | | ovncvr2.l |
. . . . . . . 8
⊢ 𝐿 = (ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘))) |
| 120 | 119 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐿 = (ℎ ∈ ((ℝ × ℝ)
↑m 𝑋)
↦ ∏𝑘 ∈
𝑋 (vol‘(([,) ∘
ℎ)‘𝑘)))) |
| 121 | | coeq2 5869 |
. . . . . . . . . . . . 13
⊢ (ℎ = (𝐼‘𝑗) → ([,) ∘ ℎ) = ([,) ∘ (𝐼‘𝑗))) |
| 122 | 121 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (ℎ = (𝐼‘𝑗) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 123 | 122 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 124 | 123 | adantllr 719 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
| 125 | 99 | adantlr 715 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘)))) |
| 126 | 114 | adantlr 715 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 127 | 124, 125,
126 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ ℎ)‘𝑘) = (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) |
| 128 | 127 | fveq2d 6910 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ ℎ)‘𝑘)) = (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))) |
| 129 | 128 | prodeq2dv 15958 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ℎ = (𝐼‘𝑗)) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ ℎ)‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))) |
| 130 | 63 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin) |
| 131 | 69 | fvmpt2 7027 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))) ∈ V) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))) |
| 132 | 53, 102, 131 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗) = (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘)))) |
| 133 | 132 | feq1d 6720 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐵‘𝑗):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ (1st ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 134 | 60, 133 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐵‘𝑗):𝑋⟶ℝ) |
| 135 | 134 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑗):𝑋⟶ℝ) |
| 136 | 135, 98 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑗)‘𝑘) ∈ ℝ) |
| 137 | 81 | fvmpt2 7027 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈ ℕ ∧ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))) ∈ V) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))) |
| 138 | 53, 109, 137 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) = (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘)))) |
| 139 | 138 | feq1d 6720 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑇‘𝑗):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ (2nd ‘((𝐼‘𝑗)‘𝑘))):𝑋⟶ℝ)) |
| 140 | 75, 139 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗):𝑋⟶ℝ) |
| 141 | 140 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝑇‘𝑗):𝑋⟶ℝ) |
| 142 | 141, 98 | ffvelcdmd 7105 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑗)‘𝑘) ∈ ℝ) |
| 143 | | volicore 46596 |
. . . . . . . . 9
⊢ ((((𝐵‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝑇‘𝑗)‘𝑘) ∈ ℝ) → (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ) |
| 144 | 136, 142,
143 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ) |
| 145 | 130, 144 | fprodrecl 15989 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∈ ℝ) |
| 146 | 120, 129,
54, 145 | fvmptd 7023 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘(𝐼‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))) |
| 147 | 146 | eqcomd 2743 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) = (𝐿‘(𝐼‘𝑗))) |
| 148 | 147 | mpteq2dva 5242 |
. . . 4
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘)))) = (𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) |
| 149 | 148 | fveq2d 6910 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗))))) |
| 150 | 47 | simprd 495 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝐿‘(𝐼‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
| 151 | 149, 150 | eqbrtrd 5165 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸)) |
| 152 | 85, 118, 151 | jca31 514 |
1
⊢ (𝜑 → (((𝐵:ℕ⟶(ℝ ↑m
𝑋) ∧ 𝑇:ℕ⟶(ℝ ↑m
𝑋)) ∧ 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐵‘𝑗)‘𝑘)[,)((𝑇‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) +𝑒 𝐸))) |