Theorem atmod4i2

Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012) (Revised by Mario Carneiro, 10-Mar-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	atmod4i2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(P\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\vee }X=\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y$$

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		hllat	$$⊢K\in \mathrm{HL}\to K\in \mathrm{Lat}$$
7	6	3ad2ant1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to K\in \mathrm{Lat}$$
8		simp21	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to P\in A$$
9	1 5	atbase	$$⊢P\in A\to P\in B$$
10	8 9	syl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to P\in B$$
11		simp23	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to Y\in B$$
12	1 4	latmcl	$$⊢\left(K\in \mathrm{Lat}\wedge P\in B\wedge Y\in B\right)\to P\underset{\dot{}}{\wedge }Y\in B$$
13	7 10 11 12	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to P\underset{\dot{}}{\wedge }Y\in B$$
14		simp22	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\in B$$
15	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge P\underset{\dot{}}{\wedge }Y\in B\wedge X\in B\right)\to \left(P\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\vee }X=X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\wedge }Y\right)$$
16	7 13 14 15	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(P\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\vee }X=X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\wedge }Y\right)$$
17	1 2 3 4 5	atmod1i2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\underset{\dot{}}{\vee }\left(P\underset{\dot{}}{\wedge }Y\right)=\left(X\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }Y$$
18	1 3	latjcom	$$⊢\left(K\in \mathrm{Lat}\wedge X\in B\wedge P\in B\right)\to X\underset{\dot{}}{\vee }P=P\underset{\dot{}}{\vee }X$$
19	7 14 10 18	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to X\underset{\dot{}}{\vee }P=P\underset{\dot{}}{\vee }X$$
20	19	oveq1d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(X\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\wedge }Y=\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y$$
21	16 17 20	3eqtrd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge X\underset{\dot{}}{\le }Y\right)\to \left(P\underset{\dot{}}{\wedge }Y\right)\underset{\dot{}}{\vee }X=\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y$$