# Metamath Proof Explorer

## 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 ${⊢}{\phi }\to ¬{u}\in {N}\left(\left\{{v}\right\}\right)$
hdmaprnlem1.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
hdmaprnlem1.q ${⊢}{Q}={0}_{{C}}$
hdmaprnlem1.o
hdmaprnlem1.a
hdmaprnlem1.t2
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 ${⊢}{\phi }\to ¬{u}\in {N}\left(\left\{{v}\right\}\right)$
15 hdmaprnlem1.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
16 hdmaprnlem1.q ${⊢}{Q}={0}_{{C}}$
17 hdmaprnlem1.o
18 hdmaprnlem1.a
19 hdmaprnlem1.t2
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
25 24 19 sseldd ${⊢}{\phi }\to {t}\in {V}$