Theorem srngmulr

Description: The multiplication operation 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	srngmulr	⊢ ( · ∈ 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		mulridx	⊢ .r = Slot ( .r ‘ ndx )
4		snsstp3	⊢ { ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
5		ssun1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
6	5 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ 𝑅
7	4 6	sstri	⊢ { ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ 𝑅
8	2 3 7	strfv	⊢ ( · ∈ 𝑋 → · = ( .r ‘ 𝑅 ) )