Theorem cgrid2

Description: Identity law for congruence. (Contributed by Scott Fenton, 12-Jun-2013)

	Ref	Expression
Assertion	cgrid2	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐴 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ → 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
2		simpr1	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
3		simpr2	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
4		simpr3	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
5		cgrcom	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐴 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐴 , 𝐴 ⟩ ) )
6	1 2 2 3 4 5	syl122anc	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐴 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐴 , 𝐴 ⟩ ) )
7		3anrot	⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ↔ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) )
8		axcgrid	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐴 , 𝐴 ⟩ → 𝐵 = 𝐶 ) )
9	7 8	sylan2b	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐴 , 𝐴 ⟩ → 𝐵 = 𝐶 ) )
10	6 9	sylbid	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐴 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ → 𝐵 = 𝐶 ) )