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Metamath Proof Explorer

Theorem pjch

Description: Projection of a vector in the projection subspace. Lemma 4.4(ii) of Beran p. 111. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion pjch ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) )

Proof

Step Hyp Ref Expression
1 eleq2 ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( 𝐴 ∈ 𝐻 ↔ 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
2 fveq2 ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( projℎ ‘ 𝐻 ) = ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
3 2 fveq1d ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) )
4 3 eqeq1d ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ) )
5 1 4 bibi12d ⊢ ( 𝐻 = if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) → ( ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) ↔ ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ) ) )
6 eleq1 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) )
7 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
8 id ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
9 7 8 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
10 6 9 bibi12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( 𝐴 ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ 𝐴 ) = 𝐴 ) ↔ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
11 ifchhv ⊢ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ∈ Cℋ
12 ifhvhv0 ⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
13 11 12 pjchi ⊢ ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ↔ ( ( projℎ ‘ if ( 𝐻 ∈ Cℋ , 𝐻 , ℋ ) ) ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
14 5 10 13 dedth2h ⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) = 𝐴 ) )