Theorem cgraid

Description: Angle congruence is reflexive. Theorem 11.6 of Schwabhauser p. 96. (Contributed by Thierry Arnoux, 31-Jul-2020)

	Ref	Expression
Hypotheses	cgraid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	cgraid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	cgraid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	cgraid.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
	cgraid.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	cgraid.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	cgraid.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	cgraid.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
	cgraid.2	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
Assertion	cgraid	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ )

Proof

Step	Hyp	Ref	Expression
1		cgraid.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		cgraid.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		cgraid.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
4		cgraid.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
5		cgraid.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
6		cgraid.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
7		cgraid.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
8		cgraid.1	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )
9		cgraid.2	⊢ ( 𝜑 → 𝐵 ≠ 𝐶 )
10		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
11		eqid	⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 )
12	1 10 2 11 3 5 6 7	cgr3id	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrG ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ )
13	1 2 4 5 5 6 3 8	hlid	⊢ ( 𝜑 → 𝐴 ( 𝐾 ‘ 𝐵 ) 𝐴 )
14	9	necomd	⊢ ( 𝜑 → 𝐶 ≠ 𝐵 )
15	1 2 4 7 5 6 3 14	hlid	⊢ ( 𝜑 → 𝐶 ( 𝐾 ‘ 𝐵 ) 𝐶 )
16	1 2 4 3 5 6 7 5 6 7 5 7 12 13 15	iscgrad	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ( cgrA ‘ 𝐺 ) ⟨“ 𝐴 𝐵 𝐶 ”⟩ )