Theorem atmod1i1

Description: Version of modular law pmod1i that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-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	atmod1i1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge P\underset{\dot{}}{\le }Y\right)\to P\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y$$

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		simpl	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\right)\to K\in \mathrm{HL}$$
7		simpr2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\right)\to X\in B$$
8		simpr1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\right)\to P\in A$$
9		eqid	$$⊢\mathrm{pmap}\left(K\right)=\mathrm{pmap}\left(K\right)$$
10		eqid	$$⊢{+}_{𝑃}\left(K\right)={+}_{𝑃}\left(K\right)$$
11	1 3 5 9 10	pmapjat2	$$⊢\left(K\in \mathrm{HL}\wedge X\in B\wedge P\in A\right)\to \mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }X\right)=\mathrm{pmap}\left(K\right)\left(P\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(X\right)$$
12	6 7 8 11	syl3anc	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\right)\to \mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }X\right)=\mathrm{pmap}\left(K\right)\left(P\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(X\right)$$
13	1 5	atbase	$$⊢P\in A\to P\in B$$
14	1 2 3 4 9 10	hlmod1i	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in B\wedge X\in B\wedge Y\in B\right)\right)\to \left(\left(P\underset{\dot{}}{\le }Y\wedge \mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }X\right)=\mathrm{pmap}\left(K\right)\left(P\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(X\right)\right)\to \left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y=P\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)\right)$$
15	13 14	syl3anr1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\right)\to \left(\left(P\underset{\dot{}}{\le }Y\wedge \mathrm{pmap}\left(K\right)\left(P\underset{\dot{}}{\vee }X\right)=\mathrm{pmap}\left(K\right)\left(P\right){+}_{𝑃}\left(K\right)\mathrm{pmap}\left(K\right)\left(X\right)\right)\to \left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y=P\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)\right)$$
16	12 15	mpan2d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\right)\to \left(P\underset{\dot{}}{\le }Y\to \left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y=P\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)\right)$$
17	16	3impia	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge P\underset{\dot{}}{\le }Y\right)\to \left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y=P\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)$$
18	17	eqcomd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge X\in B\wedge Y\in B\right)\wedge P\underset{\dot{}}{\le }Y\right)\to P\underset{\dot{}}{\vee }\left(X\underset{\dot{}}{\wedge }Y\right)=\left(P\underset{\dot{}}{\vee }X\right)\underset{\dot{}}{\wedge }Y$$