Metamath Proof Explorer


Theorem mapdpglem30b

Description: Lemma for mapdpg . (Contributed by NM, 22-Mar-2015)

Ref Expression
Hypotheses mapdpg.h
|- H = ( LHyp ` K )
mapdpg.m
|- M = ( ( mapd ` K ) ` W )
mapdpg.u
|- U = ( ( DVecH ` K ) ` W )
mapdpg.v
|- V = ( Base ` U )
mapdpg.s
|- .- = ( -g ` U )
mapdpg.z
|- .0. = ( 0g ` U )
mapdpg.n
|- N = ( LSpan ` U )
mapdpg.c
|- C = ( ( LCDual ` K ) ` W )
mapdpg.f
|- F = ( Base ` C )
mapdpg.r
|- R = ( -g ` C )
mapdpg.j
|- J = ( LSpan ` C )
mapdpg.k
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdpg.x
|- ( ph -> X e. ( V \ { .0. } ) )
mapdpg.y
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdpg.g
|- ( ph -> G e. F )
mapdpg.ne
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdpg.e
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
mapdpgem25.h1
|- ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
mapdpgem25.i1
|- ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
Assertion mapdpglem30b
|- ( ph -> i =/= ( 0g ` C ) )

Proof

Step Hyp Ref Expression
1 mapdpg.h
 |-  H = ( LHyp ` K )
2 mapdpg.m
 |-  M = ( ( mapd ` K ) ` W )
3 mapdpg.u
 |-  U = ( ( DVecH ` K ) ` W )
4 mapdpg.v
 |-  V = ( Base ` U )
5 mapdpg.s
 |-  .- = ( -g ` U )
6 mapdpg.z
 |-  .0. = ( 0g ` U )
7 mapdpg.n
 |-  N = ( LSpan ` U )
8 mapdpg.c
 |-  C = ( ( LCDual ` K ) ` W )
9 mapdpg.f
 |-  F = ( Base ` C )
10 mapdpg.r
 |-  R = ( -g ` C )
11 mapdpg.j
 |-  J = ( LSpan ` C )
12 mapdpg.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
13 mapdpg.x
 |-  ( ph -> X e. ( V \ { .0. } ) )
14 mapdpg.y
 |-  ( ph -> Y e. ( V \ { .0. } ) )
15 mapdpg.g
 |-  ( ph -> G e. F )
16 mapdpg.ne
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
17 mapdpg.e
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
18 mapdpgem25.h1
 |-  ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
19 mapdpgem25.i1
 |-  ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
20 19 simprd
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) )
21 20 simpld
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { i } ) )
22 eqid
 |-  ( LSAtoms ` U ) = ( LSAtoms ` U )
23 eqid
 |-  ( LSAtoms ` C ) = ( LSAtoms ` C )
24 1 3 12 dvhlmod
 |-  ( ph -> U e. LMod )
25 4 7 6 22 24 14 lsatlspsn
 |-  ( ph -> ( N ` { Y } ) e. ( LSAtoms ` U ) )
26 1 2 3 22 8 23 12 25 mapdat
 |-  ( ph -> ( M ` ( N ` { Y } ) ) e. ( LSAtoms ` C ) )
27 21 26 eqeltrrd
 |-  ( ph -> ( J ` { i } ) e. ( LSAtoms ` C ) )
28 eqid
 |-  ( 0g ` C ) = ( 0g ` C )
29 1 8 12 lcdlmod
 |-  ( ph -> C e. LMod )
30 19 simpld
 |-  ( ph -> i e. F )
31 9 11 28 23 29 30 lsatspn0
 |-  ( ph -> ( ( J ` { i } ) e. ( LSAtoms ` C ) <-> i =/= ( 0g ` C ) ) )
32 27 31 mpbid
 |-  ( ph -> i =/= ( 0g ` C ) )