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Metamath Proof Explorer

Theorem pjoc1

Description: Projection of a vector in the orthocomplement of the projection subspace. (Contributed by NM, 6-Nov-1999) (New usage is discouraged.)

		Ref	Expression
	Assertion	pjoc1	⊢ ( ( 𝐻 ∈ C_ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0_ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		eleq2	⊢ ( 𝐻 = if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) → ( 𝐴 ∈ 𝐻 ↔ 𝐴 ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) )
2		2fveq3	⊢ ( 𝐻 = if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) → ( proj_ℎ ‘ ( ⊥ ‘ 𝐻 ) ) = ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) )
3	2	fveq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) → ( ( proj_ℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) )
4	3	eqeq1d	⊢ ( 𝐻 = if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) → ( ( ( proj_ℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0_ℎ ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0_ℎ ) )
5	1 4	bibi12d	⊢ ( 𝐻 = if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) → ( ( 𝐴 ∈ 𝐻 ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0_ℎ ) ↔ ( 𝐴 ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0_ℎ ) ) )
6		eleq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( 𝐴 ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) )
7		fveqeq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0_ℎ ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) = 0_ℎ ) )
8	6 7	bibi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( ( 𝐴 ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ 𝐴 ) = 0_ℎ ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) = 0_ℎ ) ) )
9		ifchhv	⊢ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ∈ C_ℋ
10		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ∈ ℋ
11	9 10	pjoc1i	⊢ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ∈ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ if ( 𝐻 ∈ C_ℋ , 𝐻 , ℋ ) ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) = 0_ℎ )
12	5 8 11	dedth2h	⊢ ( ( 𝐻 ∈ C_ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( proj_ℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = 0_ℎ ) )