This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem shjcom

Description: Commutative law for Hilbert lattice join of subspaces. (Contributed by NM, 22-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion shjcom ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 shjval ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
2 shjval ⊢ ( ( 𝐵 ∈ Sℋ ∧ 𝐴 ∈ Sℋ ) → ( 𝐵 ∨ℋ 𝐴 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐵 ∪ 𝐴 ) ) ) )
3 2 ancoms ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐵 ∨ℋ 𝐴 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐵 ∪ 𝐴 ) ) ) )
4 uncom ⊢ ( 𝐵 ∪ 𝐴 ) = ( 𝐴 ∪ 𝐵 )
5 4 fveq2i ⊢ ( ⊥ ‘ ( 𝐵 ∪ 𝐴 ) ) = ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) )
6 5 fveq2i ⊢ ( ⊥ ‘ ( ⊥ ‘ ( 𝐵 ∪ 𝐴 ) ) ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) )
7 3 6 eqtrdi ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐵 ∨ℋ 𝐴 ) = ( ⊥ ‘ ( ⊥ ‘ ( 𝐴 ∪ 𝐵 ) ) ) )
8 1 7 eqtr4d ⊢ ( ( 𝐴 ∈ Sℋ ∧ 𝐵 ∈ Sℋ ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐵 ∨ℋ 𝐴 ) )