Theorem srnginvl

Description: The involution function 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	srnginvl	$$⊢\underset{\dot{}}{\ast }\in X\to \underset{\dot{}}{\ast }={\mathrm{*}}_R$$

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		starvid	$$⊢{\ast }_{𝑟}=\mathrm{Slot}{\mathrm{*}}_{\mathrm{ndx}}$$
4		ssun2	$$⊢\left\{⟨{\mathrm{*}}_{\mathrm{ndx}},\underset{\dot{}}{\ast }⟩\right\}\subseteq \left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩\right\}\cup \left\{⟨{\mathrm{*}}_{\mathrm{ndx}},\underset{\dot{}}{\ast }⟩\right\}$$
5	4 1	sseqtrri	$$⊢\left\{⟨{\mathrm{*}}_{\mathrm{ndx}},\underset{\dot{}}{\ast }⟩\right\}\subseteq R$$
6	2 3 5	strfv	$$⊢\underset{\dot{}}{\ast }\in X\to \underset{\dot{}}{\ast }={\mathrm{*}}_R$$