Theorem dalem58

Description: Lemma for dath . Analogue of dalem57 for E . (Contributed by NM, 10-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem58.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem58.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem58.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem58.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem58.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
	dalem58.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
	dalem58.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
	dalem58.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
	dalem58.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
Assertion	dalem58	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to E\underset{\dot{}}{\le }B$$

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		dalem58.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem58.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem58.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem58.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem58.e	$$⊢E=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(T\underset{\dot{}}{\vee }U\right)$$
11		dalem58.g	$$⊢G=\left(c\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }S\right)$$
12		dalem58.h	$$⊢H=\left(c\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }T\right)$$
13		dalem58.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
14		dalem58.b1	$$⊢B=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
15	1 2 3 4 8 9	dalemrot	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
16	15	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
17	1 2 3 4 8 9	dalemrotyz	$$⊢\left(\phi \wedge Y=Z\right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
18	17	3adant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
19	1 2 3 4 5 8	dalemrotps	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
20	19	3adant2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
21		biid	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)\leftrightarrow \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\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)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
22		biid	$$⊢\left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)\leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
23		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
24		eqid	$$⊢\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
25		eqid	$$⊢\left(\left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)=\left(\left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)$$
26	21 2 3 4 22 6 7 23 24 10 12 13 11 25	dalem57	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\wedge \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)\right)\to E\underset{\dot{}}{\le }\left(\left(\left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\right)$$
27	16 18 20 26	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to E\underset{\dot{}}{\le }\left(\left(\left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\right)$$
28	1	dalemkehl	$$⊢\phi \to K\in \mathrm{HL}$$
29	28	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to K\in \mathrm{HL}$$
30	1 2 3 4 5 6 7 8 9 12	dalem29	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to H\in A$$
31	1 2 3 4 5 6 7 8 9 13	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$
32	1 2 3 4 5 6 7 8 9 11	dalem23	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to G\in A$$
33	3 4	hlatjrot	$$⊢\left(K\in \mathrm{HL}\wedge \left(H\in A\wedge I\in A\wedge G\in A\right)\right)\to \left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I$$
34	29 30 31 32 33	syl13anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G=\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I$$
35	1 3 4	dalemqrprot	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
36	35 8	eqtr4di	$$⊢\phi \to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=Y$$
37	36	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=Y$$
38	34 37	oveq12d	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)=\left(\left(G\underset{\dot{}}{\vee }H\right)\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\wedge }Y$$
39	38 14	eqtr4di	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(H\underset{\dot{}}{\vee }I\right)\underset{\dot{}}{\vee }G\right)\underset{\dot{}}{\wedge }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)=B$$
40	27 39	breqtrd	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to E\underset{\dot{}}{\le }B$$