Theorem lmodvsca

Description: The scalar product operation 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	lmodvsca	$$⊢\underset{\dot{}}{·}\in X\to \underset{\dot{}}{·}={\cdot }_W$$

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		vscaid	$$⊢{\cdot }_{𝑠}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		ssun2	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),F⟩\right\}\cup \left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq W$$
6	2 3 5	strfv	$$⊢\underset{\dot{}}{·}\in X\to \underset{\dot{}}{·}={\cdot }_W$$