Theorem dalem34

Description: Lemma for dath . Analogue of dalem23 for I . (Contributed by NM, 2-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	$$⊢\psi \leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }Y\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }Y\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
	dalem34.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	dalem34.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
	dalem34.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
	dalem34.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
	dalem34.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
Assertion	dalem34	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$

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		dalem34.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		dalem34.o	$$⊢O=\mathrm{LPlanes}\left(K\right)$$
8		dalem34.y	$$⊢Y=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R$$
9		dalem34.z	$$⊢Z=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }U$$
10		dalem34.i	$$⊢I=\left(c\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\wedge }\left(d\underset{\dot{}}{\vee }U\right)$$
11	1 2 3 4 8 9	dalemrot	$$⊢\phi \to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
12	11	3ad2ant1	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)$$
13	1 2 3 4 8 9	dalemrotyz	$$⊢\left(\phi \wedge Y=Z\right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
14	13	3adant3	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
15	1 2 3 4 5 8	dalemrotps	$$⊢\left(\phi \wedge \psi \right)\to \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
16	15	3adant2	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
17		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)$$
18		biid	$$⊢\left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)\leftrightarrow \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)$$
19		eqid	$$⊢\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P$$
20		eqid	$$⊢\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S$$
21	17 2 3 4 18 6 7 19 20 10	dalem29	$$⊢\left(\left(\left(\left(K\in \mathrm{HL}\wedge C\in {\mathrm{Base}}_K\right)\wedge \left(Q\in A\wedge R\in A\wedge P\in A\right)\wedge \left(T\in A\wedge U\in A\wedge S\in A\right)\right)\wedge \left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\in O\wedge \left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\in O\right)\wedge \left(\left(\neg C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }R\right)\wedge \neg C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }P\right)\wedge \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg C\underset{\dot{}}{\le }\left(T\underset{\dot{}}{\vee }U\right)\wedge \neg C\underset{\dot{}}{\le }\left(U\underset{\dot{}}{\vee }S\right)\wedge \neg C\underset{\dot{}}{\le }\left(S\underset{\dot{}}{\vee }T\right)\right)\wedge \left(C\underset{\dot{}}{\le }\left(Q\underset{\dot{}}{\vee }T\right)\wedge C\underset{\dot{}}{\le }\left(R\underset{\dot{}}{\vee }U\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)\right)\right)\right)\wedge \left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\vee }S\wedge \left(\left(c\in A\wedge d\in A\right)\wedge \neg c\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge \left(d\ne c\wedge \neg d\underset{\dot{}}{\le }\left(\left(Q\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }P\right)\wedge C\underset{\dot{}}{\le }\left(c\underset{\dot{}}{\vee }d\right)\right)\right)\right)\to I\in A$$
22	12 14 16 21	syl3anc	$$⊢\left(\phi \wedge Y=Z\wedge \psi \right)\to I\in A$$