Theorem legeq

Description: Deduce equality from "less than" null segments. (Contributed by Thierry Arnoux, 12-Aug-2019)

	Ref	Expression
Hypotheses	legval.p	$$⊢P={\mathrm{Base}}_G$$
	legval.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	legval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	legval.l	$$⊢\underset{\dot{}}{\le }={\le }_{𝒢}\left(G\right)$$
	legval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	legid.a	$$⊢\phi \to A\in P$$
	legid.b	$$⊢\phi \to B\in P$$
	legtrd.c	$$⊢\phi \to C\in P$$
	legtrd.d	$$⊢\phi \to D\in P$$
	legeq.1	$$⊢\phi \to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}C\right)$$
Assertion	legeq	$$⊢\phi \to A=B$$

Proof

Step	Hyp	Ref	Expression
1		legval.p	$$⊢P={\mathrm{Base}}_G$$
2		legval.d	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		legval.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		legval.l	$$⊢\underset{\dot{}}{\le }={\le }_{𝒢}\left(G\right)$$
5		legval.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		legid.a	$$⊢\phi \to A\in P$$
7		legid.b	$$⊢\phi \to B\in P$$
8		legtrd.c	$$⊢\phi \to C\in P$$
9		legtrd.d	$$⊢\phi \to D\in P$$
10		legeq.1	$$⊢\phi \to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}C\right)$$
11	1 2 3 4 5 8 6 6 7	leg0	$$⊢\phi \to \left(C\underset{\dot{}}{-}C\right)\underset{\dot{}}{\le }\left(A\underset{\dot{}}{-}B\right)$$
12	1 2 3 4 5 6 7 8 8 10 11	legtri3	$$⊢\phi \to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}C$$
13	1 2 3 5 6 7 8 12	axtgcgrid	$$⊢\phi \to A=B$$