Theorem lcfrlem17

Description: Lemma for lcfr . Condition needed more than once. (Contributed by NM, 11-Mar-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	lcfrlem17.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
	lcfrlem17.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	lcfrlem17.v	$$⊢V={\mathrm{Base}}_U$$
	lcfrlem17.p	$$⊢\underset{\dot{}}{+}={+}_U$$
	lcfrlem17.z	$$⊢\underset{\dot{}}{0}=0_U$$
	lcfrlem17.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	lcfrlem17.a	$$⊢A=\mathrm{LSAtoms}\left(U\right)$$
	lcfrlem17.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	lcfrlem17.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lcfrlem17.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
	lcfrlem17.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
Assertion	lcfrlem17	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		lcfrlem17.o	$$⊢\underset{\dot{}}{\perp }=\mathrm{ocH}\left(K\right)\left(W\right)$$
3		lcfrlem17.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
4		lcfrlem17.v	$$⊢V={\mathrm{Base}}_U$$
5		lcfrlem17.p	$$⊢\underset{\dot{}}{+}={+}_U$$
6		lcfrlem17.z	$$⊢\underset{\dot{}}{0}=0_U$$
7		lcfrlem17.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
8		lcfrlem17.a	$$⊢A=\mathrm{LSAtoms}\left(U\right)$$
9		lcfrlem17.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
10		lcfrlem17.x	$$⊢\phi \to X\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
11		lcfrlem17.y	$$⊢\phi \to Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
12		lcfrlem17.ne	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
13	1 3 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
14	10	eldifad	$$⊢\phi \to X\in V$$
15	11	eldifad	$$⊢\phi \to Y\in V$$
16	4 5	lmodvacl	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{+}Y\in V$$
17	13 14 15 16	syl3anc	$$⊢\phi \to X\underset{\dot{}}{+}Y\in V$$
18	4 5 6 7 13 14 15 12	lmodindp1	$$⊢\phi \to X\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}$$
19		eldifsn	$$⊢X\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(X\underset{\dot{}}{+}Y\in V\wedge X\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}\right)$$
20	17 18 19	sylanbrc	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$