Metamath Proof Explorer

Theorem pjadji

Description: A projection is self-adjoint. Property (i) of Beran p. 109. (Contributed by NM, 6-Oct-2000) (New usage is discouraged.)

		Ref	Expression
	Hypothesis	pjadjt.1	⊢ 𝐻 ∈ C_ℋ
	Assertion	pjadji	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐵 ) = ( 𝐴 ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjadjt.1	⊢ 𝐻 ∈ C_ℋ
2		fveq2	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) )
3	2	oveq1d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐵 ) = ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih 𝐵 ) )
4		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( 𝐴 ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )
5	3 4	eqeq12d	⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) → ( ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐵 ) = ( 𝐴 ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) ↔ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) ) )
6		oveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih 𝐵 ) = ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) )
7		fveq2	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) = ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) )
8	7	oveq2d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) ) )
9	6 8	eqeq12d	⊢ ( 𝐵 = if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) → ( ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih 𝐵 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) ↔ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) ) ) )
10		ifhvhv0	⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ∈ ℋ
11		ifhvhv0	⊢ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ∈ ℋ
12	1 10 11	pjadjii	⊢ ( ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ) ·_ih if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0_ℎ ) ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ if ( 𝐵 ∈ ℋ , 𝐵 , 0_ℎ ) ) )
13	5 9 12	dedth2h	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐴 ) ·_ih 𝐵 ) = ( 𝐴 ·_ih ( ( proj_ℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )