Theorem hlexch4N

Description: A Hilbert lattice has the exchange property. Part of Definition 7.8 of MaedaMaeda p. 32. (Contributed by NM, 15-Nov-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlexch3.b	$$⊢B={\mathrm{Base}}_K$$
	hlexch3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
	hlexch3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	hlexch3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
	hlexch3.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
	hlexch3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	hlexch4N	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\wedge P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)$$

Proof

Step	Hyp	Ref	Expression
1		hlexch3.b	$$⊢B={\mathrm{Base}}_K$$
2		hlexch3.l	$$⊢\underset{\dot{}}{\le }={\le }_K$$
3		hlexch3.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
4		hlexch3.m	$$⊢\underset{\dot{}}{\wedge }=\mathrm{meet}\left(K\right)$$
5		hlexch3.z	$$⊢\underset{\dot{}}{0}=\mathrm{0.}\left(K\right)$$
6		hlexch3.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
7		hlcvl	$$⊢K\in \mathrm{HL}\to K\in \mathrm{CvLat}$$
8	1 2 3 4 5 6	cvlexch4N	$$⊢\left(K\in \mathrm{CvLat}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\wedge P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)$$
9	7 8	syl3an1	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge X\in B\right)\wedge P\underset{\dot{}}{\wedge }X=\underset{\dot{}}{0}\right)\to \left(P\underset{\dot{}}{\le }\left(X\underset{\dot{}}{\vee }Q\right)\leftrightarrow X\underset{\dot{}}{\vee }P=X\underset{\dot{}}{\vee }Q\right)$$