# Metamath Proof Explorer

## Theorem mapdh6eN

Description: Lemmma for mapdh6N . Part (6) in Baer p. 47 line 38. (Contributed by NM, 1-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q ${⊢}{Q}={0}_{{C}}$
mapdh.i
mapdh.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
mapdh.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
mapdh.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
mapdh.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
mapdh.s
mapdhc.o
mapdh.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
mapdh.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
mapdh.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
mapdh.r ${⊢}{R}={-}_{{C}}$
mapdh.j ${⊢}{J}=\mathrm{LSpan}\left({C}\right)$
mapdh.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
mapdhc.f ${⊢}{\phi }\to {F}\in {D}$
mapdh.mn ${⊢}{\phi }\to {M}\left({N}\left(\left\{{X}\right\}\right)\right)={J}\left(\left\{{F}\right\}\right)$
mapdhcl.x
mapdh.p
mapdh.a
mapdh6d.xn ${⊢}{\phi }\to ¬{X}\in {N}\left(\left\{{Y},{Z}\right\}\right)$
mapdh6d.yz ${⊢}{\phi }\to {N}\left(\left\{{Y}\right\}\right)={N}\left(\left\{{Z}\right\}\right)$
mapdh6d.y
mapdh6d.z
mapdh6d.w
mapdh6d.wn ${⊢}{\phi }\to ¬{w}\in {N}\left(\left\{{X},{Y}\right\}\right)$
Assertion mapdh6eN

### Proof

Step Hyp Ref Expression
1 mapdh.q ${⊢}{Q}={0}_{{C}}$
2 mapdh.i
3 mapdh.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
4 mapdh.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
5 mapdh.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
6 mapdh.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
7 mapdh.s
8 mapdhc.o
9 mapdh.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
10 mapdh.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
11 mapdh.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
12 mapdh.r ${⊢}{R}={-}_{{C}}$
13 mapdh.j ${⊢}{J}=\mathrm{LSpan}\left({C}\right)$
14 mapdh.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
15 mapdhc.f ${⊢}{\phi }\to {F}\in {D}$
16 mapdh.mn ${⊢}{\phi }\to {M}\left({N}\left(\left\{{X}\right\}\right)\right)={J}\left(\left\{{F}\right\}\right)$
17 mapdhcl.x
18 mapdh.p
19 mapdh.a
20 mapdh6d.xn ${⊢}{\phi }\to ¬{X}\in {N}\left(\left\{{Y},{Z}\right\}\right)$
21 mapdh6d.yz ${⊢}{\phi }\to {N}\left(\left\{{Y}\right\}\right)={N}\left(\left\{{Z}\right\}\right)$
22 mapdh6d.y
23 mapdh6d.z
24 mapdh6d.w
25 mapdh6d.wn ${⊢}{\phi }\to ¬{w}\in {N}\left(\left\{{X},{Y}\right\}\right)$
26 3 5 14 dvhlmod ${⊢}{\phi }\to {U}\in \mathrm{LMod}$
27 24 eldifad ${⊢}{\phi }\to {w}\in {V}$
28 22 eldifad ${⊢}{\phi }\to {Y}\in {V}$
29 6 18 lmodvacl
30 26 27 28 29 syl3anc
31 3 5 14 dvhlvec ${⊢}{\phi }\to {U}\in \mathrm{LVec}$
32 17 eldifad ${⊢}{\phi }\to {X}\in {V}$
33 6 9 31 27 32 28 25 lspindpi ${⊢}{\phi }\to \left({N}\left(\left\{{w}\right\}\right)\ne {N}\left(\left\{{X}\right\}\right)\wedge {N}\left(\left\{{w}\right\}\right)\ne {N}\left(\left\{{Y}\right\}\right)\right)$
34 33 simprd ${⊢}{\phi }\to {N}\left(\left\{{w}\right\}\right)\ne {N}\left(\left\{{Y}\right\}\right)$
35 6 18 8 9 26 27 28 34 lmodindp1
36 eldifsn
37 30 35 36 sylanbrc
38 23 eldifad ${⊢}{\phi }\to {Z}\in {V}$
39 6 9 31 32 28 38 20 lspindpi ${⊢}{\phi }\to \left({N}\left(\left\{{X}\right\}\right)\ne {N}\left(\left\{{Y}\right\}\right)\wedge {N}\left(\left\{{X}\right\}\right)\ne {N}\left(\left\{{Z}\right\}\right)\right)$
40 39 simpld ${⊢}{\phi }\to {N}\left(\left\{{X}\right\}\right)\ne {N}\left(\left\{{Y}\right\}\right)$
41 6 18 8 9 31 17 22 23 24 21 40 25 mapdindp3
42 6 18 8 9 31 17 22 23 24 21 40 25 mapdindp4
43 6 8 9 31 17 30 38 41 42 lspindp1
44 43 simprd
45 prcom
46 45 fveq2i
47 46 eleq2i
48 44 47 sylnibr
49 6 9 31 38 32 30 42 lspindpi
50 49 simprd
51 50 necomd
52 eqidd
53 eqidd ${⊢}{\phi }\to {I}\left(⟨{X},{F},{Z}⟩\right)={I}\left(⟨{X},{F},{Z}⟩\right)$
54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 37 23 48 51 52 53 mapdh6aN