Theorem pjnorm

Description: The norm of the projection is less than or equal to the norm. (Contributed by NM, 28-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	pjnorm	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
2	1	fveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) )
3	2	fveq2d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) )
4	3	breq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ) )
5		2fveq3	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) = ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
6		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
7	5 6	breq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ≤ ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
8		ifchhv	⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ∈ Cℋ
9		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
10	8 9	pjnormi	⊢ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ≤ ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
11	4 7 10	dedth2h	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) )