Metamath Proof Explorer


Theorem cgrcom

Description: Congruence commutes between the two sides. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion cgrcom ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ) )

Proof

Step Hyp Ref Expression
1 cgrcomim ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ → ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ) )
2 cgrcomim ( ( 𝑁 ∈ ℕ ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
3 2 3com23 ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
4 1 3 impbid ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐶 , 𝐷 ⟩ Cgr ⟨ 𝐴 , 𝐵 ⟩ ) )