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Metamath Proof Explorer

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 ) ) ) )