Theorem pjidmi

Description: A projection is idempotent. Property (ii) of Beran p. 109. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjidm.1	⊢ 𝐻 ∈ Cℋ
	pjidm.2	⊢ 𝐴 ∈ ℋ
Assertion	pjidmi	⊢ ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		pjidm.1	⊢ 𝐻 ∈ Cℋ
2		pjidm.2	⊢ 𝐴 ∈ ℋ
3	1 2	pjclii	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻
4	1 2	pjhclii	⊢ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ
5	1 4	pjchi	⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) )
6	3 5	mpbi	⊢ ( ( projℎ ‘ 𝐻 ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 )