Theorem cdleme17b

Description: Lemma leading to cdleme17c . (Contributed by NM, 11-Oct-2012)

	Ref	Expression
Hypotheses	cdleme17.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme17.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme17.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme17.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme17.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme17.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme17.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme17.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme17b	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdleme17.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme17.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme17.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme17.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme17.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme17.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
8		cdleme17.g	$$⊢G=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }\left(F\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme17.c	$$⊢C=\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		simp33	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
11		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
12		simpl1l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to K\in \mathrm{HL}$$
13	12	hllatd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to K\in \mathrm{Lat}$$
14		simpl32	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to S\in A$$
15	11 4	atbase	$$⊢S\in A\to S\in {\mathrm{Base}}_K$$
16	14 15	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to S\in {\mathrm{Base}}_K$$
17		simpl2l	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\in A$$
18	11 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
19	12 17 14 18	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
20		simpl31	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to Q\in A$$
21	11 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
22	12 17 20 21	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K$$
23	1 2 4	hlatlej2	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge S\in A\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
24	12 17 14 23	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }S\right)$$
25		simpl1r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to W\in H$$
26		simpl2r	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \neg P\underset{\dot{}}{\le }W$$
27	1 2 3 4 5 9	cdleme8	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge S\in A\right)\to P\underset{\dot{}}{\vee }C=P\underset{\dot{}}{\vee }S$$
28	12 25 17 26 14 27	syl221anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\underset{\dot{}}{\vee }C=P\underset{\dot{}}{\vee }S$$
29	1 2 4	hlatlej1	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
30	12 17 20 29	syl3anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
31		simpr	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
32	11 4	atbase	$$⊢P\in A\to P\in {\mathrm{Base}}_K$$
33	17 32	syl	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to P\in {\mathrm{Base}}_K$$
34	11 2 3 4 5 9	cdleme9b	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge S\in A\wedge W\in H\right)\right)\to C\in {\mathrm{Base}}_K$$
35	12 17 14 25 34	syl13anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to C\in {\mathrm{Base}}_K$$
36	11 1 2	latjle12	$$⊢\left(K\in \mathrm{Lat}\wedge \left(P\in {\mathrm{Base}}_K\wedge C\in {\mathrm{Base}}_K\wedge P\underset{\dot{}}{\vee }Q\in {\mathrm{Base}}_K\right)\right)\to \left(\left(P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
37	13 33 35 22 36	syl13anc	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(\left(P\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\leftrightarrow \left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
38	30 31 37	mpbi2and	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(P\underset{\dot{}}{\vee }C\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
39	28 38	eqbrtrrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
40	11 1 13 16 19 22 24 39	lattrd	$$⊢\left(\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\wedge C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\to S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
41	10 40	mtand	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(P\in A\wedge \neg P\underset{\dot{}}{\le }W\right)\wedge \left(Q\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg C\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$