Theorem llnmod2i2

Description: Version of modular law pmod1i that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join P .\/ Q ). (Contributed by NM, 16-Sep-2012) (Revised by Mario Carneiro, 10-May-2013)

	Ref	Expression
Hypotheses	atmod.b	$$⊢B={\mathrm{Base}}_K$$
	atmod.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	atmod.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	atmod.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	atmod.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	llnmod2i2	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to \left(X\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\vee }Y=X\underset{\dot{}}{\wedge }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }Y\right)$$

Proof

Step	Hyp	Ref	Expression
1		atmod.b	$$⊢B={\mathrm{Base}}_K$$
2		atmod.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		atmod.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		atmod.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		atmod.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
6		simp11	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to K\in \mathrm{HL}$$
7	6	hllatd	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to K\in \mathrm{Lat}$$
8		simp13	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Y\in B$$
9		simp2l	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to P\in A$$
10		simp2r	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Q\in A$$
11	1 3 5	hlatjcl	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge Q\in A\right)\to P\underset{\dot{}}{\vee }Q\in B$$
12	6 9 10 11	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to P\underset{\dot{}}{\vee }Q\in B$$
13		simp12	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to X\in B$$
14	1 4	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in B\wedge X\in B\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\in B$$
15	7 12 13 14	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\in B$$
16	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in B\wedge \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\in B\right)\to Y\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)=\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)\underset{\dot{}}{\vee }Y$$
17	7 8 15 16	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Y\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)=\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)\underset{\dot{}}{\vee }Y$$
18	1 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge Y\in B\wedge P\underset{\dot{}}{\vee }Q\in B\right)\to Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\in B$$
19	7 8 12 18	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\in B$$
20	1 4	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\in B\right)\to X\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)=\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }X$$
21	7 13 19 20	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to X\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)=\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }X$$
22	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\vee }Q\in B\wedge Y\in B\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }Y=Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)$$
23	7 12 8 22	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }Y=Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)$$
24	23	oveq2d	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to X\underset{\dot{}}{\wedge }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }Y\right)=X\underset{\dot{}}{\wedge }\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)$$
25		simp3	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Y\underset{\dot{}}{\le }X$$
26	1 2 3 4 5	llnmod1i2	$$⊢\left(\left(K\in \mathrm{HL}\wedge Y\in B\wedge X\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Y\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)=\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }X$$
27	6 8 13 9 10 25 26	syl321anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to Y\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)=\left(Y\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }X$$
28	21 24 27	3eqtr4d	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to X\underset{\dot{}}{\wedge }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }Y\right)=Y\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)$$
29	1 4	latmcom	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge P\underset{\dot{}}{\vee }Q\in B\right)\to X\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X$$
30	7 13 12 29	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to X\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)=\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X$$
31	30	oveq1d	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to \left(X\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\vee }Y=\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }X\right)\underset{\dot{}}{\vee }Y$$
32	17 28 31	3eqtr4rd	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge Y\underset{\dot{}}{\le }X\right)\to \left(X\underset{\dot{}}{\wedge }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\vee }Y=X\underset{\dot{}}{\wedge }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }Y\right)$$