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Metamath Proof Explorer

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 ) )