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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 → norm op ⁡ T = 0 ↔ T = 0 hop

Proof

Step Hyp Ref Expression
1 fveqeq2 ⊢ T = if T ∈ LinOp T 0 hop → norm op ⁡ T = 0 ↔ norm op ⁡ if T ∈ LinOp T 0 hop = 0
2 eqeq1 ⊢ T = if T ∈ LinOp T 0 hop → T = 0 hop ↔ if T ∈ LinOp T 0 hop = 0 hop
3 1 2 bibi12d ⊢ T = if T ∈ LinOp T 0 hop → norm op ⁡ T = 0 ↔ T = 0 hop ↔ norm op ⁡ if T ∈ LinOp T 0 hop = 0 ↔ if T ∈ LinOp T 0 hop = 0 hop
4 0lnop ⊢ 0 hop ∈ LinOp
5 4 elimel ⊢ if T ∈ LinOp T 0 hop ∈ LinOp
6 5 nmlnop0iHIL ⊢ norm op ⁡ if T ∈ LinOp T 0 hop = 0 ↔ if T ∈ LinOp T 0 hop = 0 hop
7 3 6 dedth ⊢ T ∈ LinOp → norm op ⁡ T = 0 ↔ T = 0 hop