Theorem pjcoi

Description: Composition of projections. (Contributed by NM, 16-Aug-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	⊢ 𝐺 ∈ Cℋ
	pjco.2	⊢ 𝐻 ∈ Cℋ
Assertion	pjcoi	⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ 𝐺 ) ∘ ( projℎ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ 𝐺 ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjco.1	⊢ 𝐺 ∈ Cℋ
2		pjco.2	⊢ 𝐻 ∈ Cℋ
3	1	pjfi	⊢ ( projℎ ‘ 𝐺 ) : ℋ ⟶ ℋ
4	2	pjfi	⊢ ( projℎ ‘ 𝐻 ) : ℋ ⟶ ℋ
5	3 4	hocoi	⊢ ( 𝐴 ∈ ℋ → ( ( ( projℎ ‘ 𝐺 ) ∘ ( projℎ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ 𝐺 ) ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )