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Metamath Proof Explorer

Theorem normsq

Description: The square of a norm. (Contributed by NM, 12-May-2005) (New usage is discouraged.)

Ref Expression
Assertion normsq ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) )

Proof

Step Hyp Ref Expression
1 fveq2 ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( normℎ ‘ 𝐴 ) = ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
2 1 oveq1d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) )
3 id ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
4 3 3 oveq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( 𝐴 ·ih 𝐴 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) )
5 2 4 eqeq12d ⊢ ( 𝐴 = if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) → ( ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) ↔ ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ) )
6 ifhvhv0 ⊢ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ∈ ℋ
7 6 normsqi ⊢ ( ( normℎ ‘ if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ) ↑ 2 ) = ( if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) ·ih if ( 𝐴 ∈ ℋ , 𝐴 , 0ℎ ) )
8 5 7 dedth ⊢ ( 𝐴 ∈ ℋ → ( ( normℎ ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 ·ih 𝐴 ) )