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

Metamath Proof Explorer

Theorem cmncom

Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015)

Ref Expression
Hypotheses ablcom.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
ablcom.p ⊢ + = ( +g ‘ 𝐺 )
Assertion cmncom ( ( 𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )

Proof

Step Hyp Ref Expression
1 ablcom.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 ablcom.p ⊢ + = ( +g ‘ 𝐺 )
3 1 2 iscmn ⊢ ( 𝐺 ∈ CMnd ↔ ( 𝐺 ∈ Mnd ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
4 3 simprbi ⊢ ( 𝐺 ∈ CMnd → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
5 rsp2 ⊢ ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
6 5 imp ⊢ ( ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
7 4 6 sylan ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) )
8 7 caovcomg ⊢ ( ( 𝐺 ∈ CMnd ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )
9 8 3impb ⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )