Theorem srngmulr

Description: The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015)

	Ref	Expression
Hypothesis	srngstr.r	$$⊢R=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},\underset{\dot{}}{\ast }⟩\right\}$$
Assertion	srngmulr	$$⊢\underset{\dot{}}{·}\in X\to \underset{\dot{}}{·}={\cdot }_R$$

Proof

Step	Hyp	Ref	Expression
1		srngstr.r	$$⊢R=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},\underset{\dot{}}{\ast }⟩\right\}$$
2	1	srngstr	$$⊢R\mathrm{Struct}⟨1,4⟩$$
3		mulridx	$$⊢{\cdot }_{𝑟}=\mathrm{Slot}{\cdot }_{\mathrm{ndx}}$$
4		snsstp3	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}$$
5		ssun1	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},\underset{\dot{}}{\ast }⟩\right\}$$
6	5 1	sseqtrri	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq R$$
7	4 6	sstri	$$⊢\left\{⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\subseteq R$$
8	2 3 7	strfv	$$⊢\underset{\dot{}}{·}\in X\to \underset{\dot{}}{·}={\cdot }_R$$