# Metamath Proof Explorer

## Theorem hdmaprnlem17N

Description: Lemma for hdmaprnN . Include zero. (Contributed by NM, 30-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprnlem15.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
hdmaprnlem15.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
hdmaprnlem15.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
hdmaprnlem15.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
hdmaprnlem15.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
hdmaprnlem15.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
hdmaprnlem15.q
hdmaprnlem15.l ${⊢}{L}=\mathrm{LSpan}\left({C}\right)$
hdmaprnlem15.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
hdmaprnlem15.s ${⊢}{S}=\mathrm{HDMap}\left({K}\right)\left({W}\right)$
hdmaprnlem15.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
hdmaprnlem17.se ${⊢}{\phi }\to {s}\in {D}$
Assertion hdmaprnlem17N ${⊢}{\phi }\to {s}\in \mathrm{ran}{S}$

### Proof

Step Hyp Ref Expression
1 hdmaprnlem15.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
2 hdmaprnlem15.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
3 hdmaprnlem15.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
4 hdmaprnlem15.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
5 hdmaprnlem15.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
6 hdmaprnlem15.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
7 hdmaprnlem15.q
8 hdmaprnlem15.l ${⊢}{L}=\mathrm{LSpan}\left({C}\right)$
9 hdmaprnlem15.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
10 hdmaprnlem15.s ${⊢}{S}=\mathrm{HDMap}\left({K}\right)\left({W}\right)$
11 hdmaprnlem15.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
12 hdmaprnlem17.se ${⊢}{\phi }\to {s}\in {D}$
13 eleq1
15 12 anim1i
16 eldifsn
17 15 16 sylibr
18 1 2 3 4 5 6 7 8 9 10 14 17 hdmaprnlem16N
19 eqid ${⊢}{0}_{{U}}={0}_{{U}}$
20 1 2 19 5 7 10 11 hdmapval0
21 1 2 3 10 11 hdmapfnN ${⊢}{\phi }\to {S}Fn{V}$
22 1 2 11 dvhlmod ${⊢}{\phi }\to {U}\in \mathrm{LMod}$
23 3 19 lmod0vcl ${⊢}{U}\in \mathrm{LMod}\to {0}_{{U}}\in {V}$
24 22 23 syl ${⊢}{\phi }\to {0}_{{U}}\in {V}$
25 fnfvelrn ${⊢}\left({S}Fn{V}\wedge {0}_{{U}}\in {V}\right)\to {S}\left({0}_{{U}}\right)\in \mathrm{ran}{S}$
26 21 24 25 syl2anc ${⊢}{\phi }\to {S}\left({0}_{{U}}\right)\in \mathrm{ran}{S}$
27 20 26 eqeltrrd
28 13 18 27 pm2.61ne ${⊢}{\phi }\to {s}\in \mathrm{ran}{S}$