Theorem pjige0

Description: The inner product of a projection and its argument is nonnegative. (Contributed by NM, 2-Jun-2006) (New usage is discouraged.)

	Ref	Expression
Assertion	pjige0	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → 0 ≤ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) )
2	1	fveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) )
3	2	oveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) = ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) ·ih 𝐴 ) )
4	3	breq2d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( 0 ≤ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) ↔ 0 ≤ ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) ·ih 𝐴 ) ) )
5	4	imbi2d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) → ( ( 𝐴 ∈ ℋ → 0 ≤ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) ) ↔ ( 𝐴 ∈ ℋ → 0 ≤ ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) ·ih 𝐴 ) ) ) )
6		h0elch	⊢ 0ℋ ∈ Cℋ
7	6	elimel	⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ∈ Cℋ
8	7	pjige0i	⊢ ( 𝐴 ∈ ℋ → 0 ≤ ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , 0ℋ ) ) ‘ 𝐴 ) ·ih 𝐴 ) )
9	5 8	dedth	⊢ ( 𝐻 ∈ Cℋ → ( 𝐴 ∈ ℋ → 0 ≤ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) ) )
10	9	imp	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → 0 ≤ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐴 ) )