Theorem dalem4

Description: Lemma for dalemdnee . (Contributed by NM, 10-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalem3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem3.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem3.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem3.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem3.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
	dalem3.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
Assertion	dalem4	$$⊢\left(\phi \wedge D\ne T\right)\to D\ne E$$

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		dalem3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
6		dalem3.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
7		dalem3.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
8		dalem3.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
9		dalem3.d	$$⊢D=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)$$
10		dalem3.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
11	1 2 3 4	dalemswapyz	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
12	11	adantr	$$⊢\left(\phi \wedge D\ne T\right)\to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
13	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
14	1 3 4	dalempjqeb	$$⊢\phi \to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
15	1 3 4	dalemsjteb	$$⊢\phi \to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
16		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
17	16 5	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\wedge S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
18	13 14 15 17	syl3anc	$$⊢\phi \to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(S\underset{\dot{}}{\vee }T\right)=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
19	9 18	eqtrid	$$⊢\phi \to D=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
20	19	neeq1d	$$⊢\phi \to \left(D\ne T\leftrightarrow \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\ne T\right)$$
21	20	biimpa	$$⊢\left(\phi \wedge D\ne T\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\ne T$$
22		biid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)\leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)$$
23		eqid	$$⊢\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
24		eqid	$$⊢\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
25	22 2 3 4 5 6 8 7 23 24	dalem3	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\wedge \left(Z\in O\wedge Y\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }R\right)\right)\right)\right)\wedge \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\ne T\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\ne \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
26	12 21 25	syl2anc	$$⊢\left(\phi \wedge D\ne T\right)\to \left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\ne \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
27	19	adantr	$$⊢\left(\phi \wedge D\ne T\right)\to D=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)$$
28	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
29	1	dalemqea	$$⊢\phi \to Q\in A$$
30	1	dalemrea	$$⊢\phi \to R\in A$$
31	16 3 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge Q\in A\wedge R\in A\right)\to Q\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
32	28 29 30 31	syl3anc	$$⊢\phi \to Q\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K$$
33	1 3 4	dalemtjueb	$$⊢\phi \to T\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K$$
34	16 5	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge Q\underset{\dot{}}{\vee }R\in {\mathrm{Base}}_K\wedge T\underset{\dot{}}{\vee }U\in {\mathrm{Base}}_K\right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
35	13 32 33 34	syl3anc	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
36	10 35	eqtrid	$$⊢\phi \to E=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
37	36	adantr	$$⊢\left(\phi \wedge D\ne T\right)\to E=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }R\right)$$
38	26 27 37	3netr4d	$$⊢\left(\phi \wedge D\ne T\right)\to D\ne E$$