Metamath Proof Explorer

Theorem cgrtr3

Description: Transitivity law for congruence. (Contributed by Scott Fenton, 7-Oct-2013)

Ref Expression
Assertion cgrtr3 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → A B Cgr E F ∧ C D Cgr E F → A B Cgr C D

Proof

Step Hyp Ref Expression
1 simp1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → N ∈ ℕ
2 simp21 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → A ∈ 𝔼 ⁡ N
3 simp22 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → B ∈ 𝔼 ⁡ N
4 simp32 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → E ∈ 𝔼 ⁡ N
5 simp33 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → F ∈ 𝔼 ⁡ N
6 simp23 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → C ∈ 𝔼 ⁡ N
7 simp31 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → D ∈ 𝔼 ⁡ N
8 simprl ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N ∧ A B Cgr E F ∧ C D Cgr E F → A B Cgr E F
9 simprr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N ∧ A B Cgr E F ∧ C D Cgr E F → C D Cgr E F
10 1 6 7 4 5 9 cgrcomand ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N ∧ A B Cgr E F ∧ C D Cgr E F → E F Cgr C D
11 1 2 3 4 5 6 7 8 10 cgrtrand ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N ∧ A B Cgr E F ∧ C D Cgr E F → A B Cgr C D
12 11 ex ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ E ∈ 𝔼 ⁡ N ∧ F ∈ 𝔼 ⁡ N → A B Cgr E F ∧ C D Cgr E F → A B Cgr C D