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Metamath Proof Explorer

Theorem nmlnop0

Description: A linear operator with a zero norm is identically zero. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)

Ref Expression
Assertion nmlnop0 ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) = 0 ↔ 𝑇 = 0hop ) )

Proof

Step Hyp Ref Expression
1 fveqeq2 ⊢ ( 𝑇 = if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) → ( ( normop ‘ 𝑇 ) = 0 ↔ ( normop ‘ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) ) = 0 ) )
2 eqeq1 ⊢ ( 𝑇 = if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) → ( 𝑇 = 0hop ↔ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) = 0hop ) )
3 1 2 bibi12d ⊢ ( 𝑇 = if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) → ( ( ( normop ‘ 𝑇 ) = 0 ↔ 𝑇 = 0hop ) ↔ ( ( normop ‘ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) ) = 0 ↔ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) = 0hop ) ) )
4 0lnop ⊢ 0hop ∈ LinOp
5 4 elimel ⊢ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) ∈ LinOp
6 5 nmlnop0iHIL ⊢ ( ( normop ‘ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) ) = 0 ↔ if ( 𝑇 ∈ LinOp , 𝑇 , 0hop ) = 0hop )
7 3 6 dedth ⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) = 0 ↔ 𝑇 = 0hop ) )