Theorem dalem28

Description: Lemma for dath . Lemma dalem27 expressed differently. (Contributed by NM, 4-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem23.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem23.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem23.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem23.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem23.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
Assertion	dalem28	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)$$

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		dalem23.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem23.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem23.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem23.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem23.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
11	1 2 3 4 5 6 7 8 9 10	dalem27	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)$$
12	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
13	12	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
14	5	dalemccea	$$⊢\psi \to c\in A$$
15	14	3ad2ant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\in A$$
16	1	dalempea	$$⊢\phi \to P\in A$$
17	16	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\in A$$
18	1 2 3 4 5 6 7 8 9 10	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
19	1 2 3 4 5 6 7 8 9 10	dalem25	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to c\ne G$$
20	2 3 4	hlatexch1	$$⊢\left(K\in \mathrm{HL}\wedge \left(c\in A\wedge P\in A\wedge G\in A\right)\wedge c\ne G\right)\to \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\to P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)\right)$$
21	13 15 17 18 19 20	syl131anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(c\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }P\right)\to P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)\right)$$
22	11 21	mpd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to P\underset{\dot{}}{\le }\left(G\underset{\dot{}}{\vee }c\right)$$