Theorem cdleme25b

Description: Transform cdleme24 . TODO get rid of $d's on U , N (Contributed by NM, 1-Jan-2013)

	Ref	Expression
Hypotheses	cdleme24.b	$$⊢B={\mathrm{Base}}_K$$
	cdleme24.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme24.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme24.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme24.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme24.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme24.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme24.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme24.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
Assertion	cdleme25b	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \exists u\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme24.b	$$⊢B={\mathrm{Base}}_K$$
2		cdleme24.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cdleme24.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cdleme24.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cdleme24.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		cdleme24.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
7		cdleme24.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
8		cdleme24.f	$$⊢F=\left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme24.n	$$⊢N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)$$
10	1 2 3 4 5 6 7 8 9	cdleme25a	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \exists s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge N\in B\right)$$
11		eqid	$$⊢\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
12		eqid	$$⊢\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
13	1 2 3 4 5 6 7 8 9 11 12	cdleme24	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \forall s\in A\forall t\in A\left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)$$
14		breq1	$$⊢s=t\to \left(s\underset{\dot{}}{\le }W\leftrightarrow t\underset{\dot{}}{\le }W\right)$$
15	14	notbid	$$⊢s=t\to \left(\neg s\underset{\dot{}}{\le }W\leftrightarrow \neg t\underset{\dot{}}{\le }W\right)$$
16		breq1	$$⊢s=t\to \left(s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
17	16	notbid	$$⊢s=t\to \left(\neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\leftrightarrow \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
18	15 17	anbi12d	$$⊢s=t\to \left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\leftrightarrow \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)$$
19		oveq1	$$⊢s=t\to s\underset{\dot{}}{\vee }U=t\underset{\dot{}}{\vee }U$$
20		oveq2	$$⊢s=t\to P\underset{\dot{}}{\vee }s=P\underset{\dot{}}{\vee }t$$
21	20	oveq1d	$$⊢s=t\to \left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W=\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W$$
22	21	oveq2d	$$⊢s=t\to Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)=Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)$$
23	19 22	oveq12d	$$⊢s=t\to \left(s\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
24	8 23	syl5eq	$$⊢s=t\to F=\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
25		oveq2	$$⊢s=t\to R\underset{\dot{}}{\vee }s=R\underset{\dot{}}{\vee }t$$
26	25	oveq1d	$$⊢s=t\to \left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W=\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W$$
27	24 26	oveq12d	$$⊢s=t\to F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)=\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)$$
28	27	oveq2d	$$⊢s=t\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }s\right)\underset{\dot{}}{\wedge }W\right)\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
29	9 28	syl5eq	$$⊢s=t\to N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)$$
30	18 29	reusv3	$$⊢\exists s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge N\in B\right)\to \left(\forall s\in A\forall t\in A\left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\leftrightarrow \exists u\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
31	30	biimpd	$$⊢\exists s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge N\in B\right)\to \left(\forall s\in A\forall t\in A\left(\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\wedge \left(\neg t\underset{\dot{}}{\le }W\wedge \neg t\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to N=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(\left(\left(t\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\underset{\dot{}}{\vee }\left(\left(R\underset{\dot{}}{\vee }t\right)\underset{\dot{}}{\wedge }W\right)\right)\right)\to \exists u\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)\right)$$
32	10 13 31	sylc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge \neg Q\underset{\dot{}}{\le }W\right)\right)\wedge \left(R\in A\wedge \neg R\underset{\dot{}}{\le }W\right)\wedge \left(P\ne Q\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \exists u\in B\forall s\in A\left(\left(\neg s\underset{\dot{}}{\le }W\wedge \neg s\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to u=N\right)$$