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Metamath Proof Explorer

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 ) ) )