Theorem cgr3tr

Description: Transitivity law for three-place congruence. (Contributed by Thierry Arnoux, 27-Apr-2019)

	Ref	Expression
Hypotheses	tgcgrxfr.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	tgcgrxfr.m	⊢ − = ( dist ‘ 𝐺 )
	tgcgrxfr.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	tgcgrxfr.r	⊢ ∼ = ( cgrG ‘ 𝐺 )
	tgcgrxfr.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
	tgbtwnxfr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
	tgbtwnxfr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
	tgbtwnxfr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	tgbtwnxfr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
	tgbtwnxfr.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
	tgbtwnxfr.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
	tgbtwnxfr.2	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
	cgr3tr.j	⊢ ( 𝜑 → 𝐽 ∈ 𝑃 )
	cgr3tr.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑃 )
	cgr3tr.l	⊢ ( 𝜑 → 𝐿 ∈ 𝑃 )
	cgr3tr.1	⊢ ( 𝜑 → ⟨“ 𝐷 𝐸 𝐹 ”⟩ ∼ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
Assertion	cgr3tr	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )

Proof

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		tgcgrxfr.m	⊢ − = ( dist ‘ 𝐺 )
3		tgcgrxfr.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
4		tgcgrxfr.r	⊢ ∼ = ( cgrG ‘ 𝐺 )
5		tgcgrxfr.g	⊢ ( 𝜑 → 𝐺 ∈ TarskiG )
6		tgbtwnxfr.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )
7		tgbtwnxfr.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑃 )
8		tgbtwnxfr.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
9		tgbtwnxfr.d	⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
10		tgbtwnxfr.e	⊢ ( 𝜑 → 𝐸 ∈ 𝑃 )
11		tgbtwnxfr.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑃 )
12		tgbtwnxfr.2	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐷 𝐸 𝐹 ”⟩ )
13		cgr3tr.j	⊢ ( 𝜑 → 𝐽 ∈ 𝑃 )
14		cgr3tr.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑃 )
15		cgr3tr.l	⊢ ( 𝜑 → 𝐿 ∈ 𝑃 )
16		cgr3tr.1	⊢ ( 𝜑 → ⟨“ 𝐷 𝐸 𝐹 ”⟩ ∼ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )
17	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp1	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐷 − 𝐸 ) )
18	1 2 3 4 5 9 10 11 13 14 15 16	cgr3simp1	⊢ ( 𝜑 → ( 𝐷 − 𝐸 ) = ( 𝐽 − 𝐾 ) )
19	17 18	eqtrd	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) = ( 𝐽 − 𝐾 ) )
20	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp2	⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐸 − 𝐹 ) )
21	1 2 3 4 5 9 10 11 13 14 15 16	cgr3simp2	⊢ ( 𝜑 → ( 𝐸 − 𝐹 ) = ( 𝐾 − 𝐿 ) )
22	20 21	eqtrd	⊢ ( 𝜑 → ( 𝐵 − 𝐶 ) = ( 𝐾 − 𝐿 ) )
23	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp3	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐹 − 𝐷 ) )
24	1 2 3 4 5 9 10 11 13 14 15 16	cgr3simp3	⊢ ( 𝜑 → ( 𝐹 − 𝐷 ) = ( 𝐿 − 𝐽 ) )
25	23 24	eqtrd	⊢ ( 𝜑 → ( 𝐶 − 𝐴 ) = ( 𝐿 − 𝐽 ) )
26	1 2 4 5 6 7 8 13 14 15 19 22 25	trgcgr	⊢ ( 𝜑 → ⟨“ 𝐴 𝐵 𝐶 ”⟩ ∼ ⟨“ 𝐽 𝐾 𝐿 ”⟩ )