Theorem dalempnes

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-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	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	dalempnes.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalempnes.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
Assertion	dalempnes	$$⊢\phi \to P\ne S$$

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		dalempnes.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
6		dalempnes.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
7	1	dalemkelat	$$⊢\phi \to K\in \mathrm{Lat}$$
8	1 4	dalemceb	$$⊢\phi \to C\in {\mathrm{Base}}_K$$
9	1 4	dalemseb	$$⊢\phi \to S\in {\mathrm{Base}}_K$$
10	1 4	dalemteb	$$⊢\phi \to T\in {\mathrm{Base}}_K$$
11		simp321	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\wedge \left(S\in A\wedge T\in A\wedge U\in A\right)\right)\wedge \left(Y\in O\wedge Z\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\wedge \neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\wedge C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\right)\right)\right)\to \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
12	1 11	sylbi	$$⊢\phi \to \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)$$
13		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
14	13 2 3	latnlej2l	$$⊢\left(K\in \mathrm{Lat}\wedge \left(C\in {\mathrm{Base}}_K\wedge S\in {\mathrm{Base}}_K\wedge T\in {\mathrm{Base}}_K\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\to \neg C\underset{\dot{}}{\le }S$$
15	7 8 9 10 12 14	syl131anc	$$⊢\phi \to \neg C\underset{\dot{}}{\le }S$$
16	1	dalemclpjs	$$⊢\phi \to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
17		oveq1	$$⊢P=S\to P\underset{\dot{}}{\vee }S=S\underset{\dot{}}{\vee }S$$
18	17	breq2d	$$⊢P=S\to \left(C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\leftrightarrow C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }S\right)\right)$$
19	16 18	syl5ibcom	$$⊢\phi \to \left(P=S\to C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }S\right)\right)$$
20	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
21	1	dalemsea	$$⊢\phi \to S\in A$$
22	3 4	hlatjidm	$$⊢\left(K\in \mathrm{HL}\wedge S\in A\right)\to S\underset{\dot{}}{\vee }S=S$$
23	20 21 22	syl2anc	$$⊢\phi \to S\underset{\dot{}}{\vee }S=S$$
24	23	breq2d	$$⊢\phi \to \left(C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }S\right)\leftrightarrow C\underset{\dot{}}{\le }S\right)$$
25	19 24	sylibd	$$⊢\phi \to \left(P=S\to C\underset{\dot{}}{\le }S\right)$$
26	25	necon3bd	$$⊢\phi \to \left(\neg C\underset{\dot{}}{\le }S\to P\ne S\right)$$
27	15 26	mpd	$$⊢\phi \to P\ne S$$