Theorem tgsas3

Description: First congruence theorem: SAS. Theorem 11.49 of Schwabhauser p. 107. (Contributed by Thierry Arnoux, 1-Aug-2020)

	Ref	Expression
Hypotheses	tgsas.p	$$⊢P={\mathrm{Base}}_G$$
	tgsas.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgsas.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgsas.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgsas.a	$$⊢\phi \to A\in P$$
	tgsas.b	$$⊢\phi \to B\in P$$
	tgsas.c	$$⊢\phi \to C\in P$$
	tgsas.d	$$⊢\phi \to D\in P$$
	tgsas.e	$$⊢\phi \to E\in P$$
	tgsas.f	$$⊢\phi \to F\in P$$
	tgsas.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
	tgsas.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
	tgsas.3	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
	tgsas2.4	$$⊢\phi \to A\ne C$$
Assertion	tgsas3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$

Proof

Step	Hyp	Ref	Expression
1		tgsas.p	$$⊢P={\mathrm{Base}}_G$$
2		tgsas.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgsas.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgsas.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		tgsas.a	$$⊢\phi \to A\in P$$
6		tgsas.b	$$⊢\phi \to B\in P$$
7		tgsas.c	$$⊢\phi \to C\in P$$
8		tgsas.d	$$⊢\phi \to D\in P$$
9		tgsas.e	$$⊢\phi \to E\in P$$
10		tgsas.f	$$⊢\phi \to F\in P$$
11		tgsas.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
12		tgsas.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
13		tgsas.3	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
14		tgsas2.4	$$⊢\phi \to A\ne C$$
15		eqid	$$⊢{hl}_{𝒢}\left(G\right)={hl}_{𝒢}\left(G\right)$$
16		eqid	$$⊢{\sim }_{𝒢}\left(G\right)={\sim }_{𝒢}\left(G\right)$$
17	1 2 3 4 5 6 7 8 9 10 11 12 13	tgsas	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“DEF”⟩$$
18	1 2 3 16 4 5 6 7 8 9 10 17	cgr3rotl	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}\left(G\right)⟨“EFD”⟩$$
19	1 3 15 4 5 6 7 8 9 10 12	cgrane4	$$⊢\phi \to E\ne F$$
20	1 3 15 9 5 10 4 19	hlid	$$⊢\phi \to E{hl}_{𝒢}\left(G\right)\left(F\right)E$$
21	1 2 3 4 5 6 7 8 9 10 11 12 13	tgsas1	$$⊢\phi \to C\underset{\dot{}}{-}A=F\underset{\dot{}}{-}D$$
22	1 2 3 4 7 5 10 8 21	tgcgrcomlr	$$⊢\phi \to A\underset{\dot{}}{-}C=D\underset{\dot{}}{-}F$$
23	1 2 3 4 5 7 8 10 22 14	tgcgrneq	$$⊢\phi \to D\ne F$$
24	1 3 15 8 5 10 4 23	hlid	$$⊢\phi \to D{hl}_{𝒢}\left(G\right)\left(F\right)D$$
25	1 3 15 4 6 7 5 9 10 8 9 8 18 20 24	iscgrad	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$