Step | Hyp | Ref
| Expression |
1 | | df-ovoln 40751 |
. . 3
⊢ voln* =
(𝑥 ∈ Fin ↦
(𝑦 ∈ 𝒫
(ℝ ↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
)))) |
2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → voln* = (𝑥 ∈ Fin ↦ (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
))))) |
3 | | oveq2 6658 |
. . . . 5
⊢ (𝑥 = 𝑋 → (ℝ ↑𝑚
𝑥) = (ℝ
↑𝑚 𝑋)) |
4 | 3 | pweqd 4163 |
. . . 4
⊢ (𝑥 = 𝑋 → 𝒫 (ℝ
↑𝑚 𝑥) = 𝒫 (ℝ
↑𝑚 𝑋)) |
5 | | eqeq1 2626 |
. . . . 5
⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅)) |
6 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → ((ℝ × ℝ)
↑𝑚 𝑥) = ((ℝ × ℝ)
↑𝑚 𝑋)) |
7 | 6 | oveq1d 6665 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → (((ℝ × ℝ)
↑𝑚 𝑥) ↑𝑚 ℕ) =
(((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚
ℕ)) |
8 | | ixpeq1 7919 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) |
9 | 8 | iuneq2d 4547 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) |
10 | 9 | sseq2d 3633 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (𝑦 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ↔ 𝑦 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))) |
11 | | simpl 473 |
. . . . . . . . . . . . 13
⊢ ((𝑥 = 𝑋 ∧ 𝑗 ∈ ℕ) → 𝑥 = 𝑋) |
12 | 11 | prodeq1d 14651 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑋 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) |
13 | 12 | mpteq2dva 4744 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) |
14 | 13 | fveq2d 6195 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) |
15 | 14 | eqeq2d 2632 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
16 | 10, 15 | anbi12d 747 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → ((𝑦 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ (𝑦 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))) |
17 | 7, 16 | rexeqbidv 3153 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → (∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))) |
18 | 17 | rabbidv 3189 |
. . . . . 6
⊢ (𝑥 = 𝑋 → {𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}) |
19 | 18 | infeq1d 8383 |
. . . . 5
⊢ (𝑥 = 𝑋 → inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ) =
inf({𝑧 ∈
ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
)) |
20 | 5, 19 | ifbieq2d 4111 |
. . . 4
⊢ (𝑥 = 𝑋 → if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < )) =
if(𝑋 = ∅, 0,
inf({𝑧 ∈
ℝ* ∣ ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
))) |
21 | 4, 20 | mpteq12dv 4733 |
. . 3
⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ))) =
(𝑦 ∈ 𝒫
(ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
)))) |
22 | 21 | adantl 482 |
. 2
⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑥) ↦ if(𝑥 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑥) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑥 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ))) =
(𝑦 ∈ 𝒫
(ℝ ↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
)))) |
23 | | ovnval.1 |
. 2
⊢ (𝜑 → 𝑋 ∈ Fin) |
24 | | ovex 6678 |
. . . . 5
⊢ (ℝ
↑𝑚 𝑋) ∈ V |
25 | 24 | pwex 4848 |
. . . 4
⊢ 𝒫
(ℝ ↑𝑚 𝑋) ∈ V |
26 | 25 | mptex 6486 |
. . 3
⊢ (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ))) ∈
V |
27 | 26 | a1i 11 |
. 2
⊢ (𝜑 → (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, < ))) ∈
V) |
28 | 2, 22, 23, 27 | fvmptd 6288 |
1
⊢ (𝜑 → (voln*‘𝑋) = (𝑦 ∈ 𝒫 (ℝ
↑𝑚 𝑋) ↦ if(𝑋 = ∅, 0, inf({𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)(𝑦 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))}, ℝ*, <
)))) |