Theorem dalemrotyz

Description: Lemma for dath . Rotate triangles Y = P Q R and Z = S T U to allow reuse of analogous proofs. (Contributed by NM, 19-Aug-2012)

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

Proof

Step	Hyp	Ref	Expression
1		dalema.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		dalemc.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		dalemc.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		dalemc.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		dalemrot.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
6		dalemrot.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
7		simpr	$$⊢\left(\phi \wedge Y=Z\right)\to Y=Z$$
8	1 3 4	dalemqrprot	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9	5 8	eqtr4id	$$⊢\phi \to Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
10	9	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to Y=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
11	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
12	1	dalemtea	$$⊢\phi \to T\in A$$
13	1	dalemuea	$$⊢\phi \to U\in A$$
14	1	dalemsea	$$⊢\phi \to S\in A$$
15	3 4	hlatjrot	$$⊢\left(K\in \mathrm{HL}\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\to \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
16	11 12 13 14 15	syl13anc	$$⊢\phi \to \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
17	6 16	eqtr4id	$$⊢\phi \to Z=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
18	17	adantr	$$⊢\left(\phi \wedge Y=Z\right)\to Z=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
19	7 10 18	3eqtr3d	$$⊢\left(\phi \wedge Y=Z\right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$