Theorem llnmod1i2

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	llnmod1i2	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }Y\right)=\left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }Y$$

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		simpl1	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to K\in \mathrm{HL}$$
7		simpl2	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to X\in B$$
8		simprl	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to P\in A$$
9		simprr	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to Q\in A$$
10		eqid	$$⊢\mathrm{pmap}\left(K\right)=\mathrm{pmap}\left(K\right)$$
11		eqid	$$⊢{+}_{𝑃}\left(K\right)={+}_{𝑃}\left(K\right)$$
12	1 3 5 10 11	pmapjlln1	$$⊢\left(K\in \mathrm{HL}\wedge \left(X\in B\wedge P\in A\wedge Q\in A\right)\right)\to \mathrm{pmap}\left(K\right)\left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)=\mathrm{pmap}\left(K\right)\left(X\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }Q\right)$$
13	6 7 8 9 12	syl13anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to \mathrm{pmap}\left(K\right)\left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)=\mathrm{pmap}\left(K\right)\left(X\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }Q\right)$$
14	6	hllatd	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to K\in \mathrm{Lat}$$
15	1 5	atbase	$$⊢P\in A\to P\in B$$
16	8 15	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to P\in B$$
17	1 5	atbase	$$⊢Q\in A\to Q\in B$$
18	9 17	syl	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to Q\in B$$
19	1 3	latjcl	$$⊢\left(K\in \mathrm{Lat}\wedge P\in B\wedge Q\in B\right)\to P\underset{\dot{}}{\vee }Q\in B$$
20	14 16 18 19	syl3anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to P\underset{\dot{}}{\vee }Q\in B$$
21		simpl3	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to Y\in B$$
22	1 2 3 4 10 11	hlmod1i	$$⊢\left(K\in \mathrm{HL}\wedge \left(X\in B\wedge P\underset{\dot{}}{\vee }Q\in B\wedge Y\in B\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge \mathrm{pmap}\left(K\right)\left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)=\mathrm{pmap}\left(K\right)\left(X\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }Y=X\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }Y\right)\right)$$
23	6 7 20 21 22	syl13anc	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to \left(\left(X\underset{\dot{}}{\le }Y\wedge \mathrm{pmap}\left(K\right)\left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)=\mathrm{pmap}\left(K\right)\left(X\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }Q\right)\right)\to \left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }Y=X\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }Y\right)\right)$$
24	13 23	mpan2d	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\right)\to \left(X\underset{\dot{}}{\le }Y\to \left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }Y=X\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }Y\right)\right)$$
25	24	3impia	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }Y=X\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }Y\right)$$
26	25	eqcomd	$$⊢\left(\left(K\in \mathrm{HL}\wedge X\in B\wedge Y\in B\right)\wedge \left(P\in A\wedge Q\in A\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\underset{\dot{}}{\vee }\left(\left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\wedge }Y\right)=\left(X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\vee }Q\right)\right)\underset{\dot{}}{\wedge }Y$$