Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013)
Ref | Expression | ||
---|---|---|---|
Hypotheses | grpprop.b | ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) | |
grpprop.p | ⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐿 ) | ||
Assertion | grpprop | ⊢ ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpprop.b | ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) | |
2 | grpprop.p | ⊢ ( +g ‘ 𝐾 ) = ( +g ‘ 𝐿 ) | |
3 | eqidd | ⊢ ( ⊤ → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) ) | |
4 | 1 | a1i | ⊢ ( ⊤ → ( Base ‘ 𝐾 ) = ( Base ‘ 𝐿 ) ) |
5 | 2 | oveqi | ⊢ ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) |
6 | 5 | a1i | ⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ( Base ‘ 𝐾 ) ∧ 𝑦 ∈ ( Base ‘ 𝐾 ) ) ) → ( 𝑥 ( +g ‘ 𝐾 ) 𝑦 ) = ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ) |
7 | 3 4 6 | grppropd | ⊢ ( ⊤ → ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) ) |
8 | 7 | mptru | ⊢ ( 𝐾 ∈ Grp ↔ 𝐿 ∈ Grp ) |