Metamath Proof Explorer

Theorem cmnpropd

Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015)

Ref Expression
Hypotheses ablpropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
ablpropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
ablpropd.3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
Assertion cmnpropd ( 𝜑 → ( 𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd ) )

Proof

Step Hyp Ref Expression
1 ablpropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
2 ablpropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
3 ablpropd.3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) )
4 1 2 3 mndpropd ⊢ ( 𝜑 → ( 𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd ) )
5 3 oveqrspc2v ⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) )
6 3 oveqrspc2v ⊢ ( ( 𝜑 ∧ ( 𝑣 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵 ) ) → ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) )
7 6 ancom2s ⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) )
8 5 7 eqeq12d ⊢ ( ( 𝜑 ∧ ( 𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ) ) → ( ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) )
9 8 2ralbidva ⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) )
10 1 raleqdv ⊢ ( 𝜑 → ( ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) )
11 1 10 raleqbidv ⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) )
12 2 raleqdv ⊢ ( 𝜑 → ( ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ↔ ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) )
13 2 12 raleqbidv ⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐵 ∀ 𝑣 ∈ 𝐵 ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) )
14 9 11 13 3bitr3d ⊢ ( 𝜑 → ( ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ↔ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) )
15 4 14 anbi12d ⊢ ( 𝜑 → ( ( 𝐾 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) ) )
16 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
17 eqid ⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐾 )
18 16 17 iscmn ⊢ ( 𝐾 ∈ CMnd ↔ ( 𝐾 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐾 ) ∀ 𝑣 ∈ ( Base ‘ 𝐾 ) ( 𝑢 ( +g ‘ 𝐾 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐾 ) 𝑢 ) ) )
19 eqid ⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
20 eqid ⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 )
21 19 20 iscmn ⊢ ( 𝐿 ∈ CMnd ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑢 ∈ ( Base ‘ 𝐿 ) ∀ 𝑣 ∈ ( Base ‘ 𝐿 ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑣 ) = ( 𝑣 ( +g ‘ 𝐿 ) 𝑢 ) ) )
22 15 18 21 3bitr4g ⊢ ( 𝜑 → ( 𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd ) )