Metamath Proof Explorer

Theorem mapdh8aa

Description: Part of Part (8) in Baer p. 48. (Contributed by NM, 12-May-2015)

Ref Expression
Hypotheses mapdh8a.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
mapdh8a.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
mapdh8a.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
mapdh8a.s
mapdh8a.o
mapdh8a.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
mapdh8a.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
mapdh8a.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
mapdh8a.r ${⊢}{R}={-}_{{C}}$
mapdh8a.q ${⊢}{Q}={0}_{{C}}$
mapdh8a.j ${⊢}{J}=\mathrm{LSpan}\left({C}\right)$
mapdh8a.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
mapdh8a.i
mapdh8a.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
mapdh8aa.f ${⊢}{\phi }\to {F}\in {D}$
mapdh8aa.mn ${⊢}{\phi }\to {M}\left({N}\left(\left\{{X}\right\}\right)\right)={J}\left(\left\{{F}\right\}\right)$
mapdh8aa.eg ${⊢}{\phi }\to {I}\left(⟨{X},{F},{Y}⟩\right)={G}$
mapdh8aa.ee ${⊢}{\phi }\to {I}\left(⟨{X},{F},{Z}⟩\right)={E}$
mapdh8aa.x
mapdh8aa.y
mapdh8aa.z
mapdh8aa.zt ${⊢}{\phi }\to {N}\left(\left\{{Z}\right\}\right)\ne {N}\left(\left\{{T}\right\}\right)$
mapdh8aa.t
mapdh8aa.yn ${⊢}{\phi }\to ¬{Y}\in {N}\left(\left\{{Z},{T}\right\}\right)$
mapdh8aa.xn ${⊢}{\phi }\to ¬{X}\in {N}\left(\left\{{Y},{Z}\right\}\right)$
Assertion mapdh8aa ${⊢}{\phi }\to {I}\left(⟨{Y},{G},{T}⟩\right)={I}\left(⟨{Z},{E},{T}⟩\right)$

Proof

Step Hyp Ref Expression
1 mapdh8a.h ${⊢}{H}=\mathrm{LHyp}\left({K}\right)$
2 mapdh8a.u ${⊢}{U}=\mathrm{DVecH}\left({K}\right)\left({W}\right)$
3 mapdh8a.v ${⊢}{V}={\mathrm{Base}}_{{U}}$
4 mapdh8a.s
5 mapdh8a.o
6 mapdh8a.n ${⊢}{N}=\mathrm{LSpan}\left({U}\right)$
7 mapdh8a.c ${⊢}{C}=\mathrm{LCDual}\left({K}\right)\left({W}\right)$
8 mapdh8a.d ${⊢}{D}={\mathrm{Base}}_{{C}}$
9 mapdh8a.r ${⊢}{R}={-}_{{C}}$
10 mapdh8a.q ${⊢}{Q}={0}_{{C}}$
11 mapdh8a.j ${⊢}{J}=\mathrm{LSpan}\left({C}\right)$
12 mapdh8a.m ${⊢}{M}=\mathrm{mapd}\left({K}\right)\left({W}\right)$
13 mapdh8a.i
14 mapdh8a.k ${⊢}{\phi }\to \left({K}\in \mathrm{HL}\wedge {W}\in {H}\right)$
15 mapdh8aa.f ${⊢}{\phi }\to {F}\in {D}$
16 mapdh8aa.mn ${⊢}{\phi }\to {M}\left({N}\left(\left\{{X}\right\}\right)\right)={J}\left(\left\{{F}\right\}\right)$
17 mapdh8aa.eg ${⊢}{\phi }\to {I}\left(⟨{X},{F},{Y}⟩\right)={G}$
18 mapdh8aa.ee ${⊢}{\phi }\to {I}\left(⟨{X},{F},{Z}⟩\right)={E}$
19 mapdh8aa.x
20 mapdh8aa.y
21 mapdh8aa.z
22 mapdh8aa.zt ${⊢}{\phi }\to {N}\left(\left\{{Z}\right\}\right)\ne {N}\left(\left\{{T}\right\}\right)$
23 mapdh8aa.t
24 mapdh8aa.yn ${⊢}{\phi }\to ¬{Y}\in {N}\left(\left\{{Z},{T}\right\}\right)$
25 mapdh8aa.xn ${⊢}{\phi }\to ¬{X}\in {N}\left(\left\{{Y},{Z}\right\}\right)$
26 20 eldifad ${⊢}{\phi }\to {Y}\in {V}$
27 1 2 14 dvhlvec ${⊢}{\phi }\to {U}\in \mathrm{LVec}$
28 19 eldifad ${⊢}{\phi }\to {X}\in {V}$
29 21 eldifad ${⊢}{\phi }\to {Z}\in {V}$
30 3 6 27 28 26 29 25 lspindpi ${⊢}{\phi }\to \left({N}\left(\left\{{X}\right\}\right)\ne {N}\left(\left\{{Y}\right\}\right)\wedge {N}\left(\left\{{X}\right\}\right)\ne {N}\left(\left\{{Z}\right\}\right)\right)$
31 30 simpld ${⊢}{\phi }\to {N}\left(\left\{{X}\right\}\right)\ne {N}\left(\left\{{Y}\right\}\right)$
32 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 26 31 mapdhcl ${⊢}{\phi }\to {I}\left(⟨{X},{F},{Y}⟩\right)\in {D}$
33 17 32 eqeltrrd ${⊢}{\phi }\to {G}\in {D}$
34 10 13 1 12 2 3 4 5 6 7 8 9 11 14 15 16 19 20 33 31 mapdheq
35 17 34 mpbid
36 35 simpld ${⊢}{\phi }\to {M}\left({N}\left(\left\{{Y}\right\}\right)\right)={J}\left(\left\{{G}\right\}\right)$
37 23 eldifad ${⊢}{\phi }\to {T}\in {V}$
38 3 6 27 26 29 37 24 lspindpi ${⊢}{\phi }\to \left({N}\left(\left\{{Y}\right\}\right)\ne {N}\left(\left\{{Z}\right\}\right)\wedge {N}\left(\left\{{Y}\right\}\right)\ne {N}\left(\left\{{T}\right\}\right)\right)$
39 38 simpld ${⊢}{\phi }\to {N}\left(\left\{{Y}\right\}\right)\ne {N}\left(\left\{{Z}\right\}\right)$
40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 39 25 19 20 21 mapdh75d ${⊢}{\phi }\to {I}\left(⟨{Y},{G},{Z}⟩\right)={E}$
41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 33 36 40 20 21 22 23 24 mapdh8a ${⊢}{\phi }\to {I}\left(⟨{Z},{E},{T}⟩\right)={I}\left(⟨{Y},{G},{T}⟩\right)$
42 41 eqcomd ${⊢}{\phi }\to {I}\left(⟨{Y},{G},{T}⟩\right)={I}\left(⟨{Z},{E},{T}⟩\right)$