Theorem trgcgrcom

Description: Commutative 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	$$⊢\phi \to ⟨“ABC”⟩\underset{\dot{}}{\sim }⟨“DEF”⟩$$
Assertion	trgcgrcom	$$⊢\phi \to ⟨“DEF”⟩\underset{\dot{}}{\sim }⟨“ABC”⟩$$

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	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
14	13	eqcomd	$$⊢\phi \to D\underset{\dot{}}{-}E=A\underset{\dot{}}{-}B$$
15	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp2	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
16	15	eqcomd	$$⊢\phi \to E\underset{\dot{}}{-}F=B\underset{\dot{}}{-}C$$
17	1 2 3 4 5 6 7 8 9 10 11 12	cgr3simp3	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
18	17	eqcomd	$$⊢\phi \to F\underset{\dot{}}{-}D=C\underset{\dot{}}{-}A$$
19	1 2 4 5 9 10 11 6 7 8 14 16 18	trgcgr	$$⊢\phi \to ⟨“DEF”⟩\underset{\dot{}}{\sim }⟨“ABC”⟩$$