Theorem lcfrlem19

Description: Lemma for lcfr . (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	lcfrlem19	$$⊢\phi \to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)$$

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 2 3 4 5 6 7 8 9 10 11 12	lcfrlem17	$$⊢\phi \to X\underset{\dot{}}{+}Y\in \left(V\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
14	1 2 3 4 6 9 13	dochnel	$$⊢\phi \to \neg X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)$$
15	1 3 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
16	15	adantr	$$⊢\left(\phi \wedge \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)\right)\to U\in \mathrm{LMod}$$
17	10	eldifad	$$⊢\phi \to X\in V$$
18	11	eldifad	$$⊢\phi \to Y\in V$$
19	4 5	lmodvacl	$$⊢\left(U\in \mathrm{LMod}\wedge X\in V\wedge Y\in V\right)\to X\underset{\dot{}}{+}Y\in V$$
20	15 17 18 19	syl3anc	$$⊢\phi \to X\underset{\dot{}}{+}Y\in V$$
21	20	snssd	$$⊢\phi \to \left\{X\underset{\dot{}}{+}Y\right\}\subseteq V$$
22		eqid	$$⊢\mathrm{LSubSp}\left(U\right)=\mathrm{LSubSp}\left(U\right)$$
23	1 3 4 22 2	dochlss	$$⊢\left(\left(K\in \mathrm{HL}\wedge W\in H\right)\wedge \left\{X\underset{\dot{}}{+}Y\right\}\subseteq V\right)\to \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
24	9 21 23	syl2anc	$$⊢\phi \to \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
25	24	adantr	$$⊢\left(\phi \wedge \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)\right)\to \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)$$
26		simpr	$$⊢\left(\phi \wedge \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)\right)\to \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)$$
27	5 22	lssvacl	$$⊢\left(\left(U\in \mathrm{LMod}\wedge \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\in \mathrm{LSubSp}\left(U\right)\right)\wedge \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)\right)\to X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)$$
28	16 25 26 27	syl21anc	$$⊢\left(\phi \wedge \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)\right)\to X\underset{\dot{}}{+}Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)$$
29	14 28	mtand	$$⊢\phi \to \neg \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)$$
30		ianor	$$⊢\neg \left(X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\wedge Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)\leftrightarrow \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)$$
31	29 30	sylib	$$⊢\phi \to \left(\neg X\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\vee \neg Y\in \underset{\dot{}}{\perp }\left(\left\{X\underset{\dot{}}{+}Y\right\}\right)\right)$$