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Metamath Proof Explorer

Theorem pjmuli

Description: Projection of scalar product is scalar product of projection. (Contributed by NM, 26-Nov-2000) (New usage is discouraged.)

Ref Expression
Hypothesis pjadjt.1 ⊢ 𝐻 ∈ Cℋ
Assertion pjmuli ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 pjadjt.1 ⊢ 𝐻 ∈ Cℋ
2 fvoveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( projℎ ‘ 𝐻 ) ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) )
3 oveq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( 𝐴 ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )
4 2 3 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) → ( ( ( projℎ ‘ 𝐻 ) ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) ) )
5 oveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
6 5 fveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) = ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
7 fveq2 ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) = ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
8 7 oveq2d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) )
9 6 8 eqeq12d ⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ 𝐵 ) ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) ↔ ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) ) )
10 ifhvhv0 ⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ∈ ℋ
11 0cn ⊢ 0 ∈ ℂ
12 11 elimel ⊢ if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ∈ ℂ
13 1 10 12 pjmulii ⊢ ( ( projℎ ‘ 𝐻 ) ‘ ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) ) = ( if ( 𝐴 ∈ ℂ , 𝐴 , 0 ) ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0ℎ ) ) )
14 4 9 13 dedth2h ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ ( 𝐴 ·ℎ 𝐵 ) ) = ( 𝐴 ·ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )