Theorem 0bdop

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

	Ref	Expression
Assertion	0bdop	$$⊢0_{\mathrm{hop}}\in \mathrm{BndLinOp}$$

Proof

Step	Hyp	Ref	Expression
1		0lnop	$$⊢0_{\mathrm{hop}}\in \mathrm{LinOp}$$
2		nmop0	$$⊢{norm}_{\mathrm{op}}\left(0_{\mathrm{hop}}\right)=0$$
3		0ltpnf	$$⊢0<\mathrm{+\infty }$$
4	2 3	eqbrtri	$$⊢{norm}_{\mathrm{op}}\left(0_{\mathrm{hop}}\right)<\mathrm{+\infty }$$
5		elbdop	$$⊢0_{\mathrm{hop}}\in \mathrm{BndLinOp}\leftrightarrow \left(0_{\mathrm{hop}}\in \mathrm{LinOp}\wedge {norm}_{\mathrm{op}}\left(0_{\mathrm{hop}}\right)<\mathrm{+\infty }\right)$$
6	1 4 5	mpbir2an	$$⊢0_{\mathrm{hop}}\in \mathrm{BndLinOp}$$