Theorem cdlemg2k

Description: cdleme42keg with simpler hypotheses. TODO: FIX COMMENT. TODO: derive from cdlemg3a , cdlemg2fv2 , cdlemg2jOLDN , ltrnel ? (Contributed by NM, 22-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg2inv.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdlemg2inv.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
	cdlemg2j.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdlemg2j.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdlemg2j.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdlemg2j.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdlemg2j.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdlemg2k	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge F\in T\right)\to F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)=F\left(P\right)\underset{\dot{}}{\vee }U$$

Proof

Step	Hyp	Ref	Expression
1		cdlemg2inv.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		cdlemg2inv.t	$$⊢T=\mathrm{LTrn}\left(K\right)\left(W\right)$$
3		cdlemg2j.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
4		cdlemg2j.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
5		cdlemg2j.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdlemg2j.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
7		cdlemg2j.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
9		eqid	$$⊢\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W$$
10		eqid	$$⊢\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
11		eqid	$$⊢\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12		eqid	$$⊢\left(x\in {\mathrm{Base}}_K⟼if\left(\left(p\ne q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in {\mathrm{Base}}_K|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in {\mathrm{Base}}_K|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋s/t⦌\left(\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)=\left(x\in {\mathrm{Base}}_K⟼if\left(\left(p\ne q\wedge \neg x\underset{\dot{}}{\le }W\right),\left(\iota z\in {\mathrm{Base}}_K|\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge s\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)=x\right)\to z=if\left(s\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right),\left(\iota y\in {\mathrm{Base}}_K|\forall t\in A\left(\left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(p\underset{\dot{}}{\vee }q\right)\right)\to y=\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(s\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right),⦋s/t⦌\left(\left(t\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }q\right)\underset{\dot{}}{\wedge }W\right)\right)\underset{\dot{}}{\wedge }\left(q\underset{\dot{}}{\vee }\left(\left(p\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\right)\underset{\dot{}}{\vee }\left(x\underset{\dot{}}{\wedge }W\right)\right)\right),x\right)\right)$$
13	8 3 4 6 5 1 2 9 10 11 12 7	cdlemg2klem	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(\left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge F\in T\right)\to F\left(P\right)\underset{\dot{}}{\vee }F\left(Q\right)=F\left(P\right)\underset{\dot{}}{\vee }U$$