Metamath Proof Explorer


Theorem smgrpassOLD

Description: Obsolete version of sgrpass as of 3-Feb-2020. A semigroup is associative. (Contributed by FL, 2-Nov-2009) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypothesis smgrpassOLD.1 𝑋 = dom dom 𝐺
Assertion smgrpassOLD ( 𝐺 ∈ SemiGrp → ∀ 𝑥𝑋𝑦𝑋𝑧𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 smgrpassOLD.1 𝑋 = dom dom 𝐺
2 1 issmgrpOLD ( 𝐺 ∈ SemiGrp → ( 𝐺 ∈ SemiGrp ↔ ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥𝑋𝑦𝑋𝑧𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) ) )
3 simpr ( ( 𝐺 : ( 𝑋 × 𝑋 ) ⟶ 𝑋 ∧ ∀ 𝑥𝑋𝑦𝑋𝑧𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) → ∀ 𝑥𝑋𝑦𝑋𝑧𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) )
4 2 3 syl6bi ( 𝐺 ∈ SemiGrp → ( 𝐺 ∈ SemiGrp → ∀ 𝑥𝑋𝑦𝑋𝑧𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) ) )
5 4 pm2.43i ( 𝐺 ∈ SemiGrp → ∀ 𝑥𝑋𝑦𝑋𝑧𝑋 ( ( 𝑥 𝐺 𝑦 ) 𝐺 𝑧 ) = ( 𝑥 𝐺 ( 𝑦 𝐺 𝑧 ) ) )