Theorem cdleme20aN

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 14-Nov-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme19.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cdleme19.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cdleme19.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cdleme19.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
	cdleme19.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	cdleme19.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
	cdleme19.f	$$⊢F=\left(S\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme19.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
	cdleme19.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
	cdleme19.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
	cdleme20.v	$$⊢V=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
Assertion	cdleme20aN	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$

Proof

Step	Hyp	Ref	Expression
1		cdleme19.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
2		cdleme19.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
3		cdleme19.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
4		cdleme19.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
5		cdleme19.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
6		cdleme19.u	$$⊢U=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }W$$
7		cdleme19.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		cdleme19.g	$$⊢G=\left(T\underset{\dot{}}{\vee }U\right)\underset{\dot{}}{\wedge }\left(Q\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\right)$$
9		cdleme19.d	$$⊢D=\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W$$
10		cdleme19.y	$$⊢Y=\left(R\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
11		cdleme20.v	$$⊢V=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
12	11	oveq1i	$$⊢V\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }D$$
13		simp1l	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{HL}$$
14		simp1r	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to W\in H$$
15		simp22	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\in A$$
16		simp23	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }W$$
17		simp21	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\in A$$
18		simp33	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
19		simp32	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)$$
20	1 2 3 4 5 9	cdlemeda	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(R\in A\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to D\in A$$
21	13 14 15 16 17 18 19 20	syl223anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to D\in A$$
22		simp31	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to T\in A$$
23		eqid	$$⊢{\mathrm{Base}}_K={\mathrm{Base}}_K$$
24	23 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge S\in A\wedge T\in A\right)\to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
25	13 15 22 24	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K$$
26	23 5	lhpbase	$$⊢W\in H\to W\in {\mathrm{Base}}_K$$
27	14 26	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to W\in {\mathrm{Base}}_K$$
28	13	hllatd	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to K\in \mathrm{Lat}$$
29	23 2 4	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge R\in A\wedge S\in A\right)\to R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
30	13 17 15 29	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K$$
31	23 1 3	latmle2	$$⊢\left(K\in \mathrm{Lat}\wedge R\underset{\dot{}}{\vee }S\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
32	28 30 27 31	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(R\underset{\dot{}}{\vee }S\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\le }W$$
33	9 32	eqbrtrid	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to D\underset{\dot{}}{\le }W$$
34	23 1 2 3 4	atmod4i1	$$⊢\left(K\in \mathrm{HL}\wedge \left(D\in A\wedge S\underset{\dot{}}{\vee }T\in {\mathrm{Base}}_K\wedge W\in {\mathrm{Base}}_K\right)\wedge D\underset{\dot{}}{\le }W\right)\to \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
35	13 21 25 27 33 34	syl131anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
36	1 2 3 4 5 9	cdleme10	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge R\in A\wedge \left(S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\right)\to S\underset{\dot{}}{\vee }D=S\underset{\dot{}}{\vee }R$$
37	13 14 17 15 16 36	syl212anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to S\underset{\dot{}}{\vee }D=S\underset{\dot{}}{\vee }R$$
38	37	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }T=\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T$$
39	2 4	hlatj32	$$⊢\left(K\in \mathrm{HL}\wedge \left(S\in A\wedge D\in A\wedge T\in A\right)\right)\to \left(S\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }T=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }D$$
40	13 15 21 22 39	syl13anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\vee }T=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }D$$
41	38 40	eqtr3d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T=\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }D$$
42	41	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W=\left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\vee }D\right)\underset{\dot{}}{\wedge }W$$
43	35 42	eqtr4d	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to \left(\left(S\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W\right)\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$
44	12 43	syl5eq	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left(R\in A\wedge S\in A\wedge \neg S\underset{\dot{}}{\le }W\right)\wedge \left(T\in A\wedge \neg S\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\wedge R\underset{\dot{}}{\le }\left(P\underset{\dot{}}{\vee }Q\right)\right)\right)\to V\underset{\dot{}}{\vee }D=\left(\left(S\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }T\right)\underset{\dot{}}{\wedge }W$$