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Mirrors > Home > HSE Home > Th. List > df-chsup | Structured version Visualization version GIF version |
Description: Define the supremum of a set of Hilbert lattice elements. See chsupval2 31219 for its value. We actually define the supremum for an arbitrary collection of Hilbert space subsets, not just elements of the Hilbert lattice Cℋ, to allow more general theorems. Even for general subsets the supremum still a Hilbert lattice element; see hsupcl 31148. (Contributed by NM, 9-Dec-2003.) (New usage is discouraged.) |
Ref | Expression |
---|---|
df-chsup | ⊢ ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chsup 30743 | . 2 class ∨ℋ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | chba 30728 | . . . . 5 class ℋ | |
4 | 3 | cpw 4603 | . . . 4 class 𝒫 ℋ |
5 | 4 | cpw 4603 | . . 3 class 𝒫 𝒫 ℋ |
6 | 2 | cv 1533 | . . . . . 6 class 𝑥 |
7 | 6 | cuni 4908 | . . . . 5 class ∪ 𝑥 |
8 | cort 30739 | . . . . 5 class ⊥ | |
9 | 7, 8 | cfv 6548 | . . . 4 class (⊥‘∪ 𝑥) |
10 | 9, 8 | cfv 6548 | . . 3 class (⊥‘(⊥‘∪ 𝑥)) |
11 | 2, 5, 10 | cmpt 5231 | . 2 class (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) |
12 | 1, 11 | wceq 1534 | 1 wff ∨ℋ = (𝑥 ∈ 𝒫 𝒫 ℋ ↦ (⊥‘(⊥‘∪ 𝑥))) |
Colors of variables: wff setvar class |
This definition is referenced by: hsupval 31143 |
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