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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 ∈ Cℋ
Assertion pjadji ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ·ih 𝐵 ) = ( 𝐴 ·ih ( ( projℎ ‘ 𝐻 ) ‘ 𝐵 ) ) )

Proof

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