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Metamath Proof Explorer

Theorem 0bdop

Description: The identically zero operator is bounded. (Contributed by NM, 14-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion 0bdop 0hop ∈ BndLinOp

Proof

Step Hyp Ref Expression
1 0lnop ⊢ 0hop ∈ LinOp
2 nmop0 ⊢ ( normop ‘ 0hop ) = 0
3 0ltpnf ⊢ 0 < +∞
4 2 3 eqbrtri ⊢ ( normop ‘ 0hop ) < +∞
5 elbdop ⊢ ( 0hop ∈ BndLinOp ↔ ( 0hop ∈ LinOp ∧ ( normop ‘ 0hop ) < +∞ ) )
6 1 4 5 mpbir2an ⊢ 0hop ∈ BndLinOp