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 𝐺C
pjsumt.2 𝐻C
Assertion pjdsi ( ( 𝐴 ∈ ( 𝐺 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → 𝐴 = ( ( ( proj𝐺 ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 pjsumt.1 𝐺C
2 pjsumt.2 𝐻C
3 1 2 osumi ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( 𝐺 + 𝐻 ) = ( 𝐺 𝐻 ) )
4 3 fveq2d ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( proj ‘ ( 𝐺 + 𝐻 ) ) = ( proj ‘ ( 𝐺 𝐻 ) ) )
5 4 fveq1d ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( proj ‘ ( 𝐺 + 𝐻 ) ) ‘ 𝐴 ) = ( ( proj ‘ ( 𝐺 𝐻 ) ) ‘ 𝐴 ) )
6 1 2 chjcli ( 𝐺 𝐻 ) ∈ C
7 pjid ( ( ( 𝐺 𝐻 ) ∈ C𝐴 ∈ ( 𝐺 𝐻 ) ) → ( ( proj ‘ ( 𝐺 𝐻 ) ) ‘ 𝐴 ) = 𝐴 )
8 6 7 mpan ( 𝐴 ∈ ( 𝐺 𝐻 ) → ( ( proj ‘ ( 𝐺 𝐻 ) ) ‘ 𝐴 ) = 𝐴 )
9 5 8 sylan9eqr ( ( 𝐴 ∈ ( 𝐺 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( proj ‘ ( 𝐺 + 𝐻 ) ) ‘ 𝐴 ) = 𝐴 )
10 6 cheli ( 𝐴 ∈ ( 𝐺 𝐻 ) → 𝐴 ∈ ℋ )
11 1 2 pjsumi ( 𝐴 ∈ ℋ → ( 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) → ( ( proj ‘ ( 𝐺 + 𝐻 ) ) ‘ 𝐴 ) = ( ( ( proj𝐺 ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) ) )
12 11 imp ( ( 𝐴 ∈ ℋ ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( proj ‘ ( 𝐺 + 𝐻 ) ) ‘ 𝐴 ) = ( ( ( proj𝐺 ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) )
13 10 12 sylan ( ( 𝐴 ∈ ( 𝐺 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → ( ( proj ‘ ( 𝐺 + 𝐻 ) ) ‘ 𝐴 ) = ( ( ( proj𝐺 ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) )
14 9 13 eqtr3d ( ( 𝐴 ∈ ( 𝐺 𝐻 ) ∧ 𝐺 ⊆ ( ⊥ ‘ 𝐻 ) ) → 𝐴 = ( ( ( proj𝐺 ) ‘ 𝐴 ) + ( ( proj𝐻 ) ‘ 𝐴 ) ) )