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 ) ) )