Metamath Proof Explorer


Theorem 2strop

Description: The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015) Use the index-independent version 2strop1 instead. (New usage is discouraged.)

Ref Expression
Hypotheses 2str.g 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( 𝐸 ‘ ndx ) , + ⟩ }
2str.e 𝐸 = Slot 𝑁
2str.l 1 < 𝑁
2str.n 𝑁 ∈ ℕ
Assertion 2strop ( +𝑉+ = ( 𝐸𝐺 ) )

Proof

Step Hyp Ref Expression
1 2str.g 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( 𝐸 ‘ ndx ) , + ⟩ }
2 2str.e 𝐸 = Slot 𝑁
3 2str.l 1 < 𝑁
4 2str.n 𝑁 ∈ ℕ
5 1 2 3 4 2strstr 𝐺 Struct ⟨ 1 , 𝑁
6 2 4 ndxid 𝐸 = Slot ( 𝐸 ‘ ndx )
7 snsspr2 { ⟨ ( 𝐸 ‘ ndx ) , + ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( 𝐸 ‘ ndx ) , + ⟩ }
8 7 1 sseqtrri { ⟨ ( 𝐸 ‘ ndx ) , + ⟩ } ⊆ 𝐺
9 5 6 8 strfv ( +𝑉+ = ( 𝐸𝐺 ) )