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Metamath Proof Explorer

Theorem rngocl

Description: Closure of the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

Ref Expression
Hypotheses ringi.1 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
ringi.2 ⊢ 𝐻 = ( 2nd ‘ 𝑅 )
ringi.3 ⊢ 𝑋 = ran 𝐺
Assertion rngocl ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) ∈ 𝑋 )

Proof

Step Hyp Ref Expression
1 ringi.1 ⊢ 𝐺 = ( 1st ‘ 𝑅 )
2 ringi.2 ⊢ 𝐻 = ( 2nd ‘ 𝑅 )
3 ringi.3 ⊢ 𝑋 = ran 𝐺
4 1 2 3 rngosm ⊢ ( 𝑅 ∈ RingOps → 𝐻 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 )
5 fovcdm ⊢ ( ( 𝐻 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) ∈ 𝑋 )
6 4 5 syl3an1 ⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐻 𝐵 ) ∈ 𝑋 )