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Metamath Proof Explorer

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 ) )