Theorem islsati

Description: A 1-dim subspace (atom) (of a left module or left vector space) equals the span of some vector. (Contributed by NM, 1-Oct-2014)

	Ref	Expression
Hypotheses	islsati.v	$$⊢V={\mathrm{Base}}_W$$
	islsati.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
	islsati.a	$$⊢A=\mathrm{LSAtoms}\left(W\right)$$
Assertion	islsati	$$⊢\left(W\in X\wedge U\in A\right)\to \exists v\in VU=N\left(\left\{v\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		islsati.v	$$⊢V={\mathrm{Base}}_W$$
2		islsati.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
3		islsati.a	$$⊢A=\mathrm{LSAtoms}\left(W\right)$$
4		difss	$$⊢V\setminus \left\{0_W\right\}\subseteq V$$
5		eqid	$$⊢0_W=0_W$$
6	1 2 5 3	islsat	$$⊢W\in X\to \left(U\in A\leftrightarrow \exists v\in \left(V\setminus \left\{0_W\right\}\right)U=N\left(\left\{v\right\}\right)\right)$$
7	6	biimpa	$$⊢\left(W\in X\wedge U\in A\right)\to \exists v\in \left(V\setminus \left\{0_W\right\}\right)U=N\left(\left\{v\right\}\right)$$
8		ssrexv	$$⊢V\setminus \left\{0_W\right\}\subseteq V\to \left(\exists v\in \left(V\setminus \left\{0_W\right\}\right)U=N\left(\left\{v\right\}\right)\to \exists v\in VU=N\left(\left\{v\right\}\right)\right)$$
9	4 7 8	mpsyl	$$⊢\left(W\in X\wedge U\in A\right)\to \exists v\in VU=N\left(\left\{v\right\}\right)$$