Theorem lmodbase

Description: The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	lmodstr.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
Assertion	lmodbase	$$⊢B\in X\to B={\mathrm{Base}}_W$$

Proof

Step	Hyp	Ref	Expression
1		lmodstr.w	$$⊢W=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
2	1	lmodstr	$$⊢W\mathrm{Struct}⟨1,6⟩$$
3		baseid	$$⊢\mathrm{Base}=\mathrm{Slot}{\mathrm{Base}}_{\mathrm{ndx}}$$
4		snsstp1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}$$
5		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
6	5 1	sseqtrri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}\subseteq W$$
7	4 6	sstri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩\right\}\subseteq W$$
8	2 3 7	strfv	$$⊢B\in X\to B={\mathrm{Base}}_W$$