Theorem srnginvl

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

	Ref	Expression
Hypothesis	srngstr.r	⊢ 𝑅 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
Assertion	srnginvl	⊢ ( ∗ ∈ 𝑋 → ∗ = ( *𝑟 ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		srngstr.r	⊢ 𝑅 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
2	1	srngstr	⊢ 𝑅 Struct ⟨ 1 , 4 ⟩
3		starvid	⊢ *𝑟 = Slot ( *𝑟 ‘ ndx )
4		ssun2	⊢ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
5	4 1	sseqtrri	⊢ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } ⊆ 𝑅
6	2 3 5	strfv	⊢ ( ∗ ∈ 𝑋 → ∗ = ( *𝑟 ‘ 𝑅 ) )