Metamath Proof Explorer


Theorem lmhmlvec

Description: The property for modules to be vector spaces is invariant under module isomorphism. (Contributed by Steven Nguyen, 15-Aug-2023)

Ref Expression
Assertion lmhmlvec F S LMHom T S LVec T LVec

Proof

Step Hyp Ref Expression
1 lmhmlmod1 F S LMHom T S LMod
2 lmhmlmod2 F S LMHom T T LMod
3 1 2 2thd F S LMHom T S LMod T LMod
4 eqid Scalar S = Scalar S
5 eqid Scalar T = Scalar T
6 4 5 lmhmsca F S LMHom T Scalar T = Scalar S
7 6 eqcomd F S LMHom T Scalar S = Scalar T
8 7 eleq1d F S LMHom T Scalar S DivRing Scalar T DivRing
9 3 8 anbi12d F S LMHom T S LMod Scalar S DivRing T LMod Scalar T DivRing
10 4 islvec S LVec S LMod Scalar S DivRing
11 5 islvec T LVec T LMod Scalar T DivRing
12 9 10 11 3bitr4g F S LMHom T S LVec T LVec