Theorem cvlexch4N

Description: An atomic covering lattice has the exchange property. Part of Definition 7.8 of MaedaMaeda p. 32. (Contributed by NM, 5-Nov-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cvlexch3.b	$$⊢B={\mathrm{Base}}_K$$
	cvlexch3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	cvlexch3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	cvlexch3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	cvlexch3.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
	cvlexch3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	cvlexch4N	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\wedge P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		cvlexch3.b	$$⊢B={\mathrm{Base}}_K$$
2		cvlexch3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		cvlexch3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		cvlexch3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		cvlexch3.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
6		cvlexch3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
7		cvlatl	$$⊢K\in \mathrm{CvLat}\to K\in \mathrm{AtLat}$$
8	7	adantr	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\right)\to K\in \mathrm{AtLat}$$
9		simpr1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\right)\to P\in A$$
10		simpr3	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\right)\to X\in B$$
11	1 2 4 5 6	atnle	$$⊢\left(K\in \mathrm{AtLat}\wedge P\in A\wedge X\in B\right)\to \left(\neg P\underset{\dot{}}{\le }X\leftrightarrow P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)$$
12	8 9 10 11	syl3anc	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\right)\to \left(\neg P\underset{\dot{}}{\le }X\leftrightarrow P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)$$
13	1 2 3 6	cvlexchb1	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\wedge \neg P\underset{\dot{}}{\le }X\right)\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)$$
14	13	3expia	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\right)\to \left(\neg P\underset{\dot{}}{\le }X\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)\right)$$
15	12 14	sylbird	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\right)\to \left(P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)\right)$$
16	15	3impia	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\wedge P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)$$