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Metamath Proof Explorer

Theorem grppropd

Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 2-Oct-2015)

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

Proof

Step Hyp Ref Expression
1 grppropd.1 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐾 ) )
2 grppropd.2 ⊢ ( 𝜑 → 𝐵 = ( Base ‘ 𝐿 ) )
3 grppropd.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 1 2 3 grpidpropd ⊢ ( 𝜑 → ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐿 ) )
6 5 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐿 ) )
7 3 6 eqeq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
8 7 anass1rs ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
9 8 rexbidva ⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
10 9 ralbidva ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
11 1 rexeqdv ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) )
12 1 11 raleqbidv ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) )
13 2 rexeqdv ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ↔ ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
14 2 13 raleqbidv ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
15 10 12 14 3bitr3d ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
16 4 15 anbi12d ⊢ ( 𝜑 → ( ( 𝐾 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) ) )
17 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
18 eqid ⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐾 )
19 eqid ⊢ ( 0g ‘ 𝐾 ) = ( 0g ‘ 𝐾 )
20 17 18 19 isgrp ⊢ ( 𝐾 ∈ Grp ↔ ( 𝐾 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐾 ) ∃ 𝑥 ∈ ( Base ‘ 𝐾 ) ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 0g ‘ 𝐾 ) ) )
21 eqid ⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
22 eqid ⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 )
23 eqid ⊢ ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝐿 )
24 21 22 23 isgrp ⊢ ( 𝐿 ∈ Grp ↔ ( 𝐿 ∈ Mnd ∧ ∀ 𝑦 ∈ ( Base ‘ 𝐿 ) ∃ 𝑥 ∈ ( Base ‘ 𝐿 ) ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( 0g ‘ 𝐿 ) ) )
25 16 20 24 3bitr4g ⊢ ( 𝜑 → ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) )