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Mirrors > Home > HSE Home > Th. List > df-chj | Structured version Visualization version GIF version |
Description: Define Hilbert lattice join. See chjval 30636 for its value and chjcl 30641 for its closure law. Note that we define it over all Hilbert space subsets to allow proving more general theorems. Even for general subsets the join belongs to Cℋ; see sshjcl 30639. (Contributed by NM, 1-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
df-chj | ⊢ ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chj 30217 | . 2 class ∨ℋ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | chba 30203 | . . . 4 class ℋ | |
5 | 4 | cpw 4603 | . . 3 class 𝒫 ℋ |
6 | 2 | cv 1541 | . . . . . 6 class 𝑥 |
7 | 3 | cv 1541 | . . . . . 6 class 𝑦 |
8 | 6, 7 | cun 3947 | . . . . 5 class (𝑥 ∪ 𝑦) |
9 | cort 30214 | . . . . 5 class ⊥ | |
10 | 8, 9 | cfv 6544 | . . . 4 class (⊥‘(𝑥 ∪ 𝑦)) |
11 | 10, 9 | cfv 6544 | . . 3 class (⊥‘(⊥‘(𝑥 ∪ 𝑦))) |
12 | 2, 3, 5, 5, 11 | cmpo 7411 | . 2 class (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) |
13 | 1, 12 | wceq 1542 | 1 wff ∨ℋ = (𝑥 ∈ 𝒫 ℋ, 𝑦 ∈ 𝒫 ℋ ↦ (⊥‘(⊥‘(𝑥 ∪ 𝑦)))) |
Colors of variables: wff setvar class |
This definition is referenced by: sshjval 30634 |
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