Theorem legov3

Description: An equivalent definition of the less-than relationship, from the strict relation. (Contributed by Thierry Arnoux, 15-Dec-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	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	legso.a	$$⊢E=\underset{\dot{}}{-}\left[P\times P\right]$$
	legso.f	$$⊢\phi \to \mathrm{Fun}\underset{\dot{}}{-}$$
	legso.l	$$⊢\underset{\dot{}}{<}=\left({\underset{\dot{}}{\le }\upharpoonright }_E\right)\setminus \mathrm{I}$$
	legso.d	$$⊢\phi \to P\times P\subseteq \mathrm{dom}\underset{\dot{}}{-}$$
	ltgov.a	$$⊢\phi \to A\in P$$
	ltgov.b	$$⊢\phi \to B\in P$$
Assertion	legov3	$$⊢\phi \to \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\leftrightarrow \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{<}\left(C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)$$

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		legso.a	$$⊢E=\underset{\dot{}}{-}\left[P\times P\right]$$
7		legso.f	$$⊢\phi \to \mathrm{Fun}\underset{\dot{}}{-}$$
8		legso.l	$$⊢\underset{\dot{}}{<}=\left({\underset{\dot{}}{\le }\upharpoonright }_E\right)\setminus \mathrm{I}$$
9		legso.d	$$⊢\phi \to P\times P\subseteq \mathrm{dom}\underset{\dot{}}{-}$$
10		ltgov.a	$$⊢\phi \to A\in P$$
11		ltgov.b	$$⊢\phi \to B\in P$$
12	1 2 3 4 5 6 7 8 9 10 11	ltgov	$$⊢\phi \to \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{<}\left(C\underset{\dot{}}{-}D\right)\leftrightarrow \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\right)$$
13	12	orbi1d	$$⊢\phi \to \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{<}\left(C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\leftrightarrow \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)$$
14		simprl	$$⊢\left(\left(\phi \wedge \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)\wedge \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\right)\to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)$$
15	1 2 3 4 5 10 11	legid	$$⊢\phi \to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(A\underset{\dot{}}{-}B\right)$$
16	15	adantr	$$⊢\left(\phi \wedge A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(A\underset{\dot{}}{-}B\right)$$
17		simpr	$$⊢\left(\phi \wedge A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
18	16 17	breqtrd	$$⊢\left(\phi \wedge A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)$$
19	18	adantlr	$$⊢\left(\left(\phi \wedge \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)\wedge A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)$$
20		simpr	$$⊢\left(\phi \wedge \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)\to \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)$$
21	14 19 20	mpjaodan	$$⊢\left(\phi \wedge \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)\to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)$$
22		simplr	$$⊢\left(\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\wedge \neg A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)$$
23		simpr	$$⊢\left(\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\wedge \neg A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to \neg A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D$$
24	23	neqned	$$⊢\left(\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\wedge \neg A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D$$
25	22 24	jca	$$⊢\left(\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\wedge \neg A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\to \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)$$
26	25	ex	$$⊢\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\to \left(\neg A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\to \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\right)$$
27	26	orrd	$$⊢\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\to \left(A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\vee \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\right)$$
28	27	orcomd	$$⊢\left(\phi \wedge \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)\to \left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)$$
29	21 28	impbida	$$⊢\phi \to \left(\left(\left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\wedge A\underset{\dot{}}{-}B\ne C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\leftrightarrow \left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\right)$$
30	13 29	bitr2d	$$⊢\phi \to \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{\le }\left(C\underset{\dot{}}{-}D\right)\leftrightarrow \left(\left(A\underset{\dot{}}{-}B\right)\underset{\dot{}}{<}\left(C\underset{\dot{}}{-}D\right)\vee A\underset{\dot{}}{-}B=C\underset{\dot{}}{-}D\right)\right)$$