Theorem cgr3tr

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

	Ref	Expression
Hypotheses	tgcgrxfr.p	$$⊢P={\mathrm{Base}}_G$$
	tgcgrxfr.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgcgrxfr.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgcgrxfr.r	$$⊢\underset{\dot{}}{\sim }={\sim }_{𝒢}\left(G\right)$$
	tgcgrxfr.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgbtwnxfr.a	$$⊢\phi \to A\in P$$
	tgbtwnxfr.b	$$⊢\phi \to B\in P$$
	tgbtwnxfr.c	$$⊢\phi \to C\in P$$
	tgbtwnxfr.d	$$⊢\phi \to D\in P$$
	tgbtwnxfr.e	$$⊢\phi \to E\in P$$
	tgbtwnxfr.f	$$⊢\phi \to F\in P$$
	tgbtwnxfr.2	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“DEF”⟩$$
	cgr3tr.j	$$⊢\phi \to J\in P$$
	cgr3tr.k	$$⊢\phi \to K\in P$$
	cgr3tr.l	$$⊢\phi \to L\in P$$
	cgr3tr.1	$$⊢\phi \to ⟨“DEF”⟩\underset{\dot{}}{\sim }⟨“JKL”⟩$$
Assertion	cgr3tr	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“JKL”⟩$$

Proof

Step	Hyp	Ref	Expression
1		tgcgrxfr.p	$$⊢P={\mathrm{Base}}_G$$
2		tgcgrxfr.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgcgrxfr.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgcgrxfr.r	$$⊢\underset{\dot{}}{\sim }={\sim }_{𝒢}\left(G\right)$$
5		tgcgrxfr.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		tgbtwnxfr.a	$$⊢\phi \to A\in P$$
7		tgbtwnxfr.b	$$⊢\phi \to B\in P$$
8		tgbtwnxfr.c	$$⊢\phi \to C\in P$$
9		tgbtwnxfr.d	$$⊢\phi \to D\in P$$
10		tgbtwnxfr.e	$$⊢\phi \to E\in P$$
11		tgbtwnxfr.f	$$⊢\phi \to F\in P$$
12		tgbtwnxfr.2	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“DEF”⟩$$
13		cgr3tr.j	$$⊢\phi \to J\in P$$
14		cgr3tr.k	$$⊢\phi \to K\in P$$
15		cgr3tr.l	$$⊢\phi \to L\in P$$
16		cgr3tr.1	$$⊢\phi \to ⟨“DEF”⟩\underset{\dot{}}{\sim }⟨“JKL”⟩$$
17	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
18	1 2 3 4 5 9 10 11 13 14 15 16	cgr3simp1	$$⊢\phi \to D\underset{\dot{}}{-}E=J\underset{\dot{}}{-}K$$
19	17 18	eqtrd	$$⊢\phi \to A\underset{\dot{}}{-}B=J\underset{\dot{}}{-}K$$
20	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp2	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
21	1 2 3 4 5 9 10 11 13 14 15 16	cgr3simp2	$$⊢\phi \to E\underset{\dot{}}{-}F=K\underset{\dot{}}{-}L$$
22	20 21	eqtrd	$$⊢\phi \to B\underset{\dot{}}{-}C=K\underset{\dot{}}{-}L$$
23	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp3	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
24	1 2 3 4 5 9 10 11 13 14 15 16	cgr3simp3	$$⊢\phi \to F\underset{\dot{}}{-}D=L\underset{\dot{}}{-}J$$
25	23 24	eqtrd	$$⊢\phi \to C\underset{\dot{}}{-}A=L\underset{\dot{}}{-}J$$
26	1 2 4 5 6 7 8 13 14 15 19 22 25	trgcgr	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“JKL”⟩$$