This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem srngbase

Description: The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Hypothesis srngstr.r ⊢ 𝑅 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
Assertion srngbase ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 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 baseid ⊢ Base = Slot ( Base ‘ ndx )
4 snsstp1 ⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( 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 ⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝑅
8 2 3 7 strfv ⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 𝑅 ) )