Theorem algmulr

Description: The multiplicative operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	algpart.a	$$⊢A=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
Assertion	algmulr	$$⊢\underset{\dot{}}{\times }\in V\to \underset{\dot{}}{\times }={\cdot }_A$$

Proof

Step	Hyp	Ref	Expression
1		algpart.a	$$⊢A=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
2	1	algstr	$$⊢A\mathrm{Struct}⟨1,6⟩$$
3		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		snsstp3	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}$$
5		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
6	5 1	sseqtrri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\subseteq A$$
7	4 6	sstri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\subseteq A$$
8	2 3 7	strfv	$$⊢\underset{\dot{}}{\times }\in V\to \underset{\dot{}}{\times }={\cdot }_A$$