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Metamath Proof Explorer

Theorem sshjval3

Description: Value of join for subsets of Hilbert space in terms of supremum: the join is the supremum of its two arguments. Based on the definition of join in Beran p. 3. For later convenience we prove a general version that works for any subset of Hilbert space, not just the elements of the lattice CH . (Contributed by NM, 2-Mar-2004) (Revised by Mario Carneiro, 23-Dec-2013) (New usage is discouraged.)

Ref Expression
Assertion sshjval3 ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ∨ℋ ‘ { 𝐴 , 𝐵 } ) )

Proof

Step Hyp Ref Expression
1 ax-hilex ⊢ ℋ ∈ V
2 1 elpw2 ⊢ ( 𝐴 ∈ 𝒫 ℋ ↔ 𝐴 ⊆ ℋ )
3 1 elpw2 ⊢ ( 𝐵 ∈ 𝒫 ℋ ↔ 𝐵 ⊆ ℋ )
4 uniprg ⊢ ( ( 𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) )
5 2 3 4 syl2anbr ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ∪ { 𝐴 , 𝐵 } = ( 𝐴 ∪ 𝐵 ) )
6 5 fveq2d ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( ⊥ ‘ ∪ { 𝐴 , 𝐵 } ) = ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) )
7 6 fveq2d ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( ⊥ ‘ ( ⊥ ‘ ∪ { 𝐴 , 𝐵 } ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
8 prssi ⊢ ( ( 𝐴 ∈ 𝒫 ℋ ∧ 𝐵 ∈ 𝒫 ℋ ) → { 𝐴 , 𝐵 } ⊆ 𝒫 ℋ )
9 2 3 8 syl2anbr ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → { 𝐴 , 𝐵 } ⊆ 𝒫 ℋ )
10 hsupval ⊢ ( { 𝐴 , 𝐵 } ⊆ 𝒫 ℋ → ( ∨ℋ ‘ { 𝐴 , 𝐵 } ) = ( ⊥ ‘ ( ⊥ ‘ ∪ { 𝐴 , 𝐵 } ) ) )
11 9 10 syl ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( ∨ℋ ‘ { 𝐴 , 𝐵 } ) = ( ⊥ ‘ ( ⊥ ‘ ∪ { 𝐴 , 𝐵 } ) ) )
12 sshjval ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
13 7 11 12 3eqtr4rd ⊢ ( ( 𝐴 ⊆ ℋ ∧ 𝐵 ⊆ ℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ∨ℋ ‘ { 𝐴 , 𝐵 } ) )