Metamath Proof Explorer


Theorem cgrdegen

Description: Two congruent segments are either both degenerate or both nondegenerate. (Contributed by Scott Fenton, 12-Jun-2013)

Ref Expression
Assertion cgrdegen ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ → ( 𝐴 = 𝐵𝐶 = 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 opeq1 ( 𝐴 = 𝐵 → ⟨ 𝐴 , 𝐵 ⟩ = ⟨ 𝐵 , 𝐵 ⟩ )
2 1 breq1d ( 𝐴 = 𝐵 → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐵 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) )
3 2 biimpac ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ 𝐴 = 𝐵 ) → ⟨ 𝐵 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
4 simp1 ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
5 simp2r ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
6 simp3l ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
7 simp3r ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
8 cgrid2 ( ( 𝑁 ∈ ℕ ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐵 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ → 𝐶 = 𝐷 ) )
9 4 5 6 7 8 syl13anc ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐵 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ → 𝐶 = 𝐷 ) )
10 3 9 syl5 ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ 𝐴 = 𝐵 ) → 𝐶 = 𝐷 ) )
11 10 expdimp ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) → ( 𝐴 = 𝐵𝐶 = 𝐷 ) )
12 opeq1 ( 𝐶 = 𝐷 → ⟨ 𝐶 , 𝐷 ⟩ = ⟨ 𝐷 , 𝐷 ⟩ )
13 12 breq2d ( 𝐶 = 𝐷 → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ↔ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐷 ⟩ ) )
14 13 biimpac ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ 𝐶 = 𝐷 ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐷 ⟩ )
15 simp2l ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
16 axcgrid ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐷 ⟩ → 𝐴 = 𝐵 ) )
17 4 15 5 7 16 syl13anc ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐷 ⟩ → 𝐴 = 𝐵 ) )
18 14 17 syl5 ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ∧ 𝐶 = 𝐷 ) → 𝐴 = 𝐵 ) )
19 18 expdimp ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) → ( 𝐶 = 𝐷𝐴 = 𝐵 ) )
20 11 19 impbid ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) → ( 𝐴 = 𝐵𝐶 = 𝐷 ) )
21 20 ex ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ → ( 𝐴 = 𝐵𝐶 = 𝐷 ) ) )