Theorem dalemrotps

Description: Lemma for dath . Rotate triangles Y = P Q R and Z = S T U to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012)

	Ref	Expression
Hypotheses	dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
	dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalemrotps.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
Assertion	dalemrotps	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		dalem.ph	$$⊢\phi \leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)$$
2		dalem.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalem.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalem.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalem.ps	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
6		dalemrotps.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7	5	dalemccea	$$⊢\psi \to c\in A$$
8	5	dalemddea	$$⊢\psi \to d\in A$$
9	7 8	jca	$$⊢\psi \to \left(c\in A\wedge d\in A\right)$$
10	9	adantl	$$⊢\left(\phi \wedge \psi \right)\to \left(c\in A\wedge d\in A\right)$$
11	5	dalem-ccly	$$⊢\psi \to \neg c\underset{\dot{}}{\le }Y$$
12	11	adantl	$$⊢\left(\phi \wedge \psi \right)\to \neg c\underset{\dot{}}{\le }Y$$
13	1 3 4	dalemqrprot	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
14	6 13	eqtr4id	$$⊢\phi \to Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
15	14	breq2d	$$⊢\phi \to \left(c\underset{\dot{}}{\le }Y\leftrightarrow c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\right)$$
16	15	adantr	$$⊢\left(\phi \wedge \psi \right)\to \left(c\underset{\dot{}}{\le }Y\leftrightarrow c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\right)$$
17	12 16	mtbid	$$⊢\left(\phi \wedge \psi \right)\to \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)$$
18	5	dalemccnedd	$$⊢\psi \to c\ne d$$
19	18	necomd	$$⊢\psi \to d\ne c$$
20	19	adantl	$$⊢\left(\phi \wedge \psi \right)\to d\ne c$$
21	5	dalem-ddly	$$⊢\psi \to \neg d\underset{\dot{}}{\le }Y$$
22	21	adantl	$$⊢\left(\phi \wedge \psi \right)\to \neg d\underset{\dot{}}{\le }Y$$
23	14	breq2d	$$⊢\phi \to \left(d\underset{\dot{}}{\le }Y\leftrightarrow d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\right)$$
24	23	adantr	$$⊢\left(\phi \wedge \psi \right)\to \left(d\underset{\dot{}}{\le }Y\leftrightarrow d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\right)$$
25	22 24	mtbid	$$⊢\left(\phi \wedge \psi \right)\to \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)$$
26	5	dalemclccjdd	$$⊢\psi \to C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)$$
27	26	adantl	$$⊢\left(\phi \wedge \psi \right)\to C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)$$
28	20 25 27	3jca	$$⊢\left(\phi \wedge \psi \right)\to \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)$$
29	10 17 28	3jca	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$