Theorem lmodindp1

Description: Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015)

	Ref	Expression
Hypotheses	lmodindp1.v	$$⊢V={\mathrm{Base}}_W$$
	lmodindp1.p	$$⊢\underset{\dot{}}{+}={+}_W$$
	lmodindp1.o	$$⊢\underset{\dot{}}{0}=0_W$$
	lmodindp1.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
	lmodindp1.w	$$⊢\phi \to W\in \mathrm{LMod}$$
	lmodindp1.x	$$⊢\phi \to X\in V$$
	lmodindp1.y	$$⊢\phi \to Y\in V$$
	lmodindp1.q	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
Assertion	lmodindp1	$$⊢\phi \to X\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}$$

Proof

Step	Hyp	Ref	Expression
1		lmodindp1.v	$$⊢V={\mathrm{Base}}_W$$
2		lmodindp1.p	$$⊢\underset{\dot{}}{+}={+}_W$$
3		lmodindp1.o	$$⊢\underset{\dot{}}{0}=0_W$$
4		lmodindp1.n	$$⊢N=\mathrm{LSpan}\left(W\right)$$
5		lmodindp1.w	$$⊢\phi \to W\in \mathrm{LMod}$$
6		lmodindp1.x	$$⊢\phi \to X\in V$$
7		lmodindp1.y	$$⊢\phi \to Y\in V$$
8		lmodindp1.q	$$⊢\phi \to N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)$$
9		eqid	$$⊢{inv}_g\left(W\right)={inv}_g\left(W\right)$$
10	1 9 4	lspsnneg	$$⊢\left(W\in \mathrm{LMod}\wedge X\in V\right)\to N\left(\left\{{inv}_g\left(W\right)\left(X\right)\right\}\right)=N\left(\left\{X\right\}\right)$$
11	5 6 10	syl2anc	$$⊢\phi \to N\left(\left\{{inv}_g\left(W\right)\left(X\right)\right\}\right)=N\left(\left\{X\right\}\right)$$
12	11	eqcomd	$$⊢\phi \to N\left(\left\{X\right\}\right)=N\left(\left\{{inv}_g\left(W\right)\left(X\right)\right\}\right)$$
13	12	adantr	$$⊢\left(\phi \wedge X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)\to N\left(\left\{X\right\}\right)=N\left(\left\{{inv}_g\left(W\right)\left(X\right)\right\}\right)$$
14		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
15	5 14	syl	$$⊢\phi \to W\in \mathrm{Grp}$$
16	1 2 3 9	grpinvid1	$$⊢\left(W\in \mathrm{Grp}\wedge X\in V\wedge Y\in V\right)\to \left({inv}_g\left(W\right)\left(X\right)=Y\leftrightarrow X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)$$
17	15 6 7 16	syl3anc	$$⊢\phi \to \left({inv}_g\left(W\right)\left(X\right)=Y\leftrightarrow X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)$$
18	17	biimpar	$$⊢\left(\phi \wedge X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)\to {inv}_g\left(W\right)\left(X\right)=Y$$
19	18	sneqd	$$⊢\left(\phi \wedge X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)\to \left\{{inv}_g\left(W\right)\left(X\right)\right\}=\left\{Y\right\}$$
20	19	fveq2d	$$⊢\left(\phi \wedge X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)\to N\left(\left\{{inv}_g\left(W\right)\left(X\right)\right\}\right)=N\left(\left\{Y\right\}\right)$$
21	13 20	eqtrd	$$⊢\left(\phi \wedge X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\right)\to N\left(\left\{X\right\}\right)=N\left(\left\{Y\right\}\right)$$
22	21	ex	$$⊢\phi \to \left(X\underset{\dot{}}{+}Y=\underset{\dot{}}{0}\to N\left(\left\{X\right\}\right)=N\left(\left\{Y\right\}\right)\right)$$
23	22	necon3d	$$⊢\phi \to \left(N\left(\left\{X\right\}\right)\ne N\left(\left\{Y\right\}\right)\to X\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}\right)$$
24	8 23	mpd	$$⊢\phi \to X\underset{\dot{}}{+}Y\ne \underset{\dot{}}{0}$$