Theorem laut1o

Description: A lattice automorphism is one-to-one and onto. (Contributed by NM, 19-May-2012)

	Ref	Expression
Hypotheses	laut1o.b	$$⊢B={\mathrm{Base}}_K$$
	laut1o.i	$$⊢I=\mathrm{LAut}\left(K\right)$$
Assertion	laut1o	$$⊢\left(K\in A\wedge F\in I\right)\to F:B\underset{1-1\; onto}{⟶}B$$

Proof

Step	Hyp	Ref	Expression
1		laut1o.b	$$⊢B={\mathrm{Base}}_K$$
2		laut1o.i	$$⊢I=\mathrm{LAut}\left(K\right)$$
3		eqid	$$⊢{\le }_K={\le }_K$$
4	1 3 2	islaut	$$⊢K\in A\to \left(F\in I\leftrightarrow \left(F:B\underset{1-1\; onto}{⟶}B\wedge \forall x\in B\forall y\in B\left(x{\le }_{K}y\leftrightarrow F\left(x\right){\le }_{K}F\left(y\right)\right)\right)\right)$$
5	4	simprbda	$$⊢\left(K\in A\wedge F\in I\right)\to F:B\underset{1-1\; onto}{⟶}B$$