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Metamath Proof Explorer

Theorem iscmn

Description: The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015)

Ref Expression
Hypotheses iscmn.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
iscmn.p ⊢ + = ( +g ‘ 𝐺 )
Assertion iscmn ( 𝐺 ∈ CMnd ↔ ( 𝐺 ∈ Mnd ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 iscmn.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 iscmn.p ⊢ + = ( +g ‘ 𝐺 )
3 fveq2 ⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = ( Base ‘ 𝐺 ) )
4 3 1 eqtr4di ⊢ ( 𝑔 = 𝐺 → ( Base ‘ 𝑔 ) = 𝐵 )
5 raleq ⊢ ( ( Base ‘ 𝑔 ) = 𝐵 → ( ∀ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ↔ ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ) )
6 5 raleqbi1dv ⊢ ( ( Base ‘ 𝑔 ) = 𝐵 → ( ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ∀ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ) )
7 4 6 syl ⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ∀ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ) )
8 fveq2 ⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = ( +g ‘ 𝐺 ) )
9 8 2 eqtr4di ⊢ ( 𝑔 = 𝐺 → ( +g ‘ 𝑔 ) = + )
10 9 oveqd ⊢ ( 𝑔 = 𝐺 → ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑥 + 𝑦 ) )
11 9 oveqd ⊢ ( 𝑔 = 𝐺 → ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) = ( 𝑦 + 𝑥 ) )
12 10 11 eqeq12d ⊢ ( 𝑔 = 𝐺 → ( ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ↔ ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
13 12 2ralbidv ⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
14 7 13 bitrd ⊢ ( 𝑔 = 𝐺 → ( ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ∀ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )
15 df-cmn ⊢ CMnd = { 𝑔 ∈ Mnd ∣ ∀ 𝑥 ∈ ( Base ‘ 𝑔 ) ∀ 𝑦 ∈ ( Base ‘ 𝑔 ) ( 𝑥 ( +g ‘ 𝑔 ) 𝑦 ) = ( 𝑦 ( +g ‘ 𝑔 ) 𝑥 ) }
16 14 15 elrab2 ⊢ ( 𝐺 ∈ CMnd ↔ ( 𝐺 ∈ Mnd ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) )