Theorem hdmaprnlem4tN

Description: Lemma for hdmaprnN . TODO: This lemma doesn't quite pay for itself even though used six times. Maybe prove this directly instead. (Contributed by NM, 27-May-2015) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hdmaprnlem1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
	hdmaprnlem1.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
	hdmaprnlem1.v	$$⊢V={\mathrm{Base}}_U$$
	hdmaprnlem1.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
	hdmaprnlem1.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
	hdmaprnlem1.l	$$⊢L=\mathrm{LSpan}\left(C\right)$$
	hdmaprnlem1.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
	hdmaprnlem1.s	$$⊢S=\mathrm{HDMap}\left(K\right)\left(W\right)$$
	hdmaprnlem1.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
	hdmaprnlem1.se	$$⊢\phi \to s\in \left(D\setminus \left\{Q\right\}\right)$$
	hdmaprnlem1.ve	$$⊢\phi \to v\in V$$
	hdmaprnlem1.e	$$⊢\phi \to M\left(N\left(\left\{v\right\}\right)\right)=L\left(\left\{s\right\}\right)$$
	hdmaprnlem1.ue	$$⊢\phi \to u\in V$$
	hdmaprnlem1.un	$$⊢\phi \to \neg u\in N\left(\left\{v\right\}\right)$$
	hdmaprnlem1.d	$$⊢D={\mathrm{Base}}_C$$
	hdmaprnlem1.q	$$⊢Q=0_C$$
	hdmaprnlem1.o	$$⊢\underset{\dot{}}{0}=0_U$$
	hdmaprnlem1.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
	hdmaprnlem1.t2	$$⊢\phi \to t\in \left(N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
Assertion	hdmaprnlem4tN	$$⊢\phi \to t\in V$$

Proof

Step	Hyp	Ref	Expression
1		hdmaprnlem1.h	$$⊢H=\mathrm{LHyp}\left(K\right)$$
2		hdmaprnlem1.u	$$⊢U=\mathrm{DVecH}\left(K\right)\left(W\right)$$
3		hdmaprnlem1.v	$$⊢V={\mathrm{Base}}_U$$
4		hdmaprnlem1.n	$$⊢N=\mathrm{LSpan}\left(U\right)$$
5		hdmaprnlem1.c	$$⊢C=\mathrm{LCDual}\left(K\right)\left(W\right)$$
6		hdmaprnlem1.l	$$⊢L=\mathrm{LSpan}\left(C\right)$$
7		hdmaprnlem1.m	$$⊢M=\mathrm{mapd}\left(K\right)\left(W\right)$$
8		hdmaprnlem1.s	$$⊢S=\mathrm{HDMap}\left(K\right)\left(W\right)$$
9		hdmaprnlem1.k	$$⊢\phi \to \left(K\in \mathrm{HL}\wedge W\in H\right)$$
10		hdmaprnlem1.se	$$⊢\phi \to s\in \left(D\setminus \left\{Q\right\}\right)$$
11		hdmaprnlem1.ve	$$⊢\phi \to v\in V$$
12		hdmaprnlem1.e	$$⊢\phi \to M\left(N\left(\left\{v\right\}\right)\right)=L\left(\left\{s\right\}\right)$$
13		hdmaprnlem1.ue	$$⊢\phi \to u\in V$$
14		hdmaprnlem1.un	$$⊢\phi \to \neg u\in N\left(\left\{v\right\}\right)$$
15		hdmaprnlem1.d	$$⊢D={\mathrm{Base}}_C$$
16		hdmaprnlem1.q	$$⊢Q=0_C$$
17		hdmaprnlem1.o	$$⊢\underset{\dot{}}{0}=0_U$$
18		hdmaprnlem1.a	$$⊢\underset{\dot{}}{✚}={+}_C$$
19		hdmaprnlem1.t2	$$⊢\phi \to t\in \left(N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\right)$$
20	1 2 9	dvhlmod	$$⊢\phi \to U\in \mathrm{LMod}$$
21	11	snssd	$$⊢\phi \to \left\{v\right\}\subseteq V$$
22	3 4	lspssv	$$⊢\left(U\in \mathrm{LMod}\wedge \left\{v\right\}\subseteq V\right)\to N\left(\left\{v\right\}\right)\subseteq V$$
23	20 21 22	syl2anc	$$⊢\phi \to N\left(\left\{v\right\}\right)\subseteq V$$
24	23	ssdifssd	$$⊢\phi \to N\left(\left\{v\right\}\right)\setminus \left\{\underset{\dot{}}{0}\right\}\subseteq V$$
25	24 19	sseldd	$$⊢\phi \to t\in V$$