Theorem pjdsi

Description: Vector decomposition into sum of projections on orthogonal subspaces. (Contributed by NM, 21-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pjsumt.1	⊢ 𝐺 ∈ Cℋ
	pjsumt.2	⊢ 𝐻 ∈ Cℋ
Assertion	pjdsi	⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 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	1 2	chjcli	⊢ ( 𝐺 ∨ℋ 𝐻 ) ∈ Cℋ
7		pjid	⊢ ( ( ( 𝐺 ∨ℋ 𝐻 ) ∈ Cℋ ∧ 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = 𝐴 )
8	6 7	mpan	⊢ ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 ∨ℋ 𝐻 ) ) ‘ 𝐴 ) = 𝐴 )
9	5 8	sylan9eqr	⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = 𝐴 )
10	6	cheli	⊢ ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) → 𝐴 ∈ ℋ )
11	1 2	pjsumi	⊢ ( 𝐴 ∈ ℋ → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ) )
12	11	imp	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )
13	10 12	sylan	⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( projℎ ‘ ( 𝐺 +ℋ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )
14	9 13	eqtr3d	⊢ ( ( 𝐴 ∈ ( 𝐺 ∨ℋ 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → 𝐴 = ( ( ( projℎ ‘ 𝐺 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )