Theorem pjssumi

Description: The projection on a subspace sum is the sum of the projections. (Contributed by NM, 11-Nov-2000) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjco.1	⊢ 𝐺 ∈ Cℋ
	pjco.2	⊢ 𝐻 ∈ Cℋ
Assertion	pjssumi	⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) = ( ( projℎ ‘ 𝐺 ) +op ( projℎ ‘ 𝐻 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjco.1	⊢ 𝐺 ∈ Cℋ
2		pjco.2	⊢ 𝐻 ∈ Cℋ
3	1 2	osumi	⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( 𝐺 +ℋ 𝐻 ) = ( 𝐺 ∨ℋ 𝐻 ) )
4	3	fveq2d	⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) = ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) )
5	1 2	pjscji	⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) = ( ( projℎ ‘ 𝐺 ) +op ( projℎ ‘ 𝐻 ) ) )
6	4 5	eqtrd	⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) = ( ( projℎ ‘ 𝐺 ) +op ( projℎ ‘ 𝐻 ) ) )