Theorem hdmaplem3

Description: Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015)

	Ref	Expression
Hypotheses	hdmaplem1.v	$$⊢V={\mathrm{Base}}_W$$
	hdmaplem1.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
	hdmaplem1.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	hdmaplem1.z	$$⊢\phi \to Z\in V$$
	hdmaplem1.j	$$⊢\phi \to \neg Z\in \left(N\left(\left\{X\right\}\right)\cup N\left(\left\{Y\right\}\right)\right)$$
	hdmaplem1.y	$$⊢\phi \to Y\in V$$
	hdmaplem3.o	$$⊢\underset{\dot{}}{0}=0_W$$
Assertion	hdmaplem3	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		hdmaplem1.v	$$⊢V={\mathrm{Base}}_W$$
2		hdmaplem1.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
3		hdmaplem1.w	$$⊢\phi \to W\in \mathrm{LMod}$$
4		hdmaplem1.z	$$⊢\phi \to Z\in V$$
5		hdmaplem1.j	$$⊢\phi \to \neg Z\in \left(N\left(\left\{X\right\}\right)\cup N\left(\left\{Y\right\}\right)\right)$$
6		hdmaplem1.y	$$⊢\phi \to Y\in V$$
7		hdmaplem3.o	$$⊢\underset{\dot{}}{0}=0_W$$
8		eqid	$$⊢\mathrm{LSubSp}\left(W\right)=\mathrm{LSubSp}\left(W\right)$$
9	1 8 2	lspsncl	$$⊢\left(W\in \mathrm{LMod}\wedge Y\in V\right)\to N\left(\left\{Y\right\}\right)\in \mathrm{LSubSp}\left(W\right)$$
10	3 6 9	syl2anc	$$⊢\phi \to N\left(\left\{Y\right\}\right)\in \mathrm{LSubSp}\left(W\right)$$
11		elun2	$$⊢Z\in N\left(\left\{Y\right\}\right)\to Z\in \left(N\left(\left\{X\right\}\right)\cup N\left(\left\{Y\right\}\right)\right)$$
12	5 11	nsyl	$$⊢\phi \to \neg Z\in N\left(\left\{Y\right\}\right)$$
13	7 8 3 10 4 12	lssneln0	$$⊢\phi \to Z\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$