Theorem pjsumi

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	pjsumt.1	⊢ 𝐺 ∈ Cℋ
	pjsumt.2	⊢ 𝐻 ∈ Cℋ
Assertion	pjsumi	⊢ ( 𝐴 ∈ ℋ → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjsumt.1	⊢ 𝐺 ∈ Cℋ
2		pjsumt.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	4	fveq1d	⊢ ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) )
6	5	adantl	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) )
7		pjcjt2	⊢ ( ( 𝐺 ∈ Cℋ ∧ 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) )
8	1 2 7	mp3an12	⊢ ( 𝐴 ∈ ℋ → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) )
9	8	imp	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )
10	6 9	eqtrd	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )
11	10	ex	⊢ ( 𝐴 ∈ ℋ → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) )