# Metamath Proof Explorer

## Theorem hdmap1l6k

Description: Lemmma for hdmap1l6 . Eliminate nonzero vector requirement. (Contributed by NM, 1-May-2015)

Ref Expression
Hypotheses hdmap1l6.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
hdmap1l6.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
hdmap1l6.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
hdmap1l6.p
hdmap1l6.s
hdmap1l6c.o
hdmap1l6.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
hdmap1l6.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
hdmap1l6.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
hdmap1l6.a
hdmap1l6.r ${⊢}{R}={-}_{{C}}$
hdmap1l6.q ${⊢}{Q}={0}_{{C}}$
hdmap1l6.l ${⊢}{L}=\mathrm{LSpan}\left({C}\right)$
hdmap1l6.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
hdmap1l6.i ${⊢}{I}=\mathrm{HDMap1}\left({K}\right)\left({W}\right)$
hdmap1l6.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
hdmap1l6.f ${⊢}{\phi }\to {F}\in {D}$
hdmap1l6cl.x
hdmap1l6.mn ${⊢}{\phi }\to {M}\left({N}\left(\left\{{X}\right\}\right)\right)={L}\left(\left\{{F}\right\}\right)$
hdmap1l6k.y ${⊢}{\phi }\to {Y}\in {V}$
hdmap1l6k.z ${⊢}{\phi }\to {Z}\in {V}$
hdmap1l6k.xn ${⊢}{\phi }\to ¬{X}\in {N}\left(\left\{{Y},{Z}\right\}\right)$
Assertion hdmap1l6k

### Proof

Step Hyp Ref Expression
1 hdmap1l6.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
2 hdmap1l6.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
3 hdmap1l6.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
4 hdmap1l6.p
5 hdmap1l6.s
6 hdmap1l6c.o
7 hdmap1l6.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
8 hdmap1l6.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
9 hdmap1l6.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
10 hdmap1l6.a
11 hdmap1l6.r ${⊢}{R}={-}_{{C}}$
12 hdmap1l6.q ${⊢}{Q}={0}_{{C}}$
13 hdmap1l6.l ${⊢}{L}=\mathrm{LSpan}\left({C}\right)$
14 hdmap1l6.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
15 hdmap1l6.i ${⊢}{I}=\mathrm{HDMap1}\left({K}\right)\left({W}\right)$
16 hdmap1l6.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
17 hdmap1l6.f ${⊢}{\phi }\to {F}\in {D}$
18 hdmap1l6cl.x
19 hdmap1l6.mn ${⊢}{\phi }\to {M}\left({N}\left(\left\{{X}\right\}\right)\right)={L}\left(\left\{{F}\right\}\right)$
20 hdmap1l6k.y ${⊢}{\phi }\to {Y}\in {V}$
21 hdmap1l6k.z ${⊢}{\phi }\to {Z}\in {V}$
22 hdmap1l6k.xn ${⊢}{\phi }\to ¬{X}\in {N}\left(\left\{{Y},{Z}\right\}\right)$
27 simpr
30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 23 24 25 26 27 28 29 hdmap1l6b
36 simpr
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 31 32 33 34 35 36 37 hdmap1l6c
45 simprl
46 eldifsn
47 44 45 46 sylanbrc
49 simprr
50 eldifsn
51 48 49 50 sylanbrc
52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 39 40 41 42 43 47 51 hdmap1l6j
53 30 38 52 pm2.61da2ne