Theorem lvecisfrlm

Description: Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019)

	Ref	Expression
Hypothesis	lvecisfrlm.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	lvecisfrlm	$$⊢W\in \mathrm{LVec}\to \exists kW{\simeq }_{𝑚}\left(F\mathrm{freeLMod}k\right)$$

Proof

Step	Hyp	Ref	Expression
1		lvecisfrlm.f	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2		eqid	$$⊢\mathrm{LBasis}\left(W\right)=\mathrm{LBasis}\left(W\right)$$
3	2	lbsex	$$⊢W\in \mathrm{LVec}\to \mathrm{LBasis}\left(W\right)\ne \varnothing$$
4		lveclmod	$$⊢W\in \mathrm{LVec}\to W\in \mathrm{LMod}$$
5	2 1	lmisfree	$$⊢W\in \mathrm{LMod}\to \left(\mathrm{LBasis}\left(W\right)\ne \varnothing \leftrightarrow \exists kW{\simeq }_{𝑚}\left(F\mathrm{freeLMod}k\right)\right)$$
6	4 5	syl	$$⊢W\in \mathrm{LVec}\to \left(\mathrm{LBasis}\left(W\right)\ne \varnothing \leftrightarrow \exists kW{\simeq }_{𝑚}\left(F\mathrm{freeLMod}k\right)\right)$$
7	3 6	mpbid	$$⊢W\in \mathrm{LVec}\to \exists kW{\simeq }_{𝑚}\left(F\mathrm{freeLMod}k\right)$$