Theorem pjnel

Description: If a vector does not belong to subspace, the norm of its projection is less than its norm. (Contributed by NM, 2-Nov-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	pjnel	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( 𝐴 ∈ 𝐻 ↔ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
2	1	notbid	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ¬ 𝐴 ∈ 𝐻 ↔ ¬ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
3		fveq2	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
4	3	fveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) )
5	4	fveq2d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) )
6	5	breq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) )
7	2 6	bibi12d	⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( ¬ 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ↔ ( ¬ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ) )
8		eleq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
9	8	notbid	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ¬ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ¬ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
10		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 ) ) ) )
11		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
12	10 11	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 ) ) ) )
13	9 12	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ¬ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ↔ ( ¬ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) < ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) ) )
14		ifchhv	⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ∈ Cℋ
15		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
16	14 15	pjneli	⊢ ( ¬ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( normℎ ‘ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) < ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
17	7 13 16	dedth2h	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) )