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 ) )