Metamath Proof Explorer

Theorem mapdpglem29

Description: Lemma for mapdpg . Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (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 ) } ) ) ) )
mapdpglem26.a 
|- A = ( Scalar ` U )
mapdpglem26.b 
|- B = ( Base ` A )
mapdpglem26.t 
|- .x. = ( .s ` C )
mapdpglem26.o 
|- O = ( 0g ` A )
mapdpglem28.ve 
|- ( ph -> v e. B )
mapdpglem28.u1 
|- ( ph -> h = ( u .x. i ) )
mapdpglem28.u2 
|- ( ph -> ( G R h ) = ( v .x. ( G R i ) ) )
Assertion mapdpglem29 
|- ( ph -> ( J ` { G } ) =/= ( J ` { i } ) )

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 mapdpglem26.a 
 |-  A = ( Scalar ` U )
21 mapdpglem26.b 
 |-  B = ( Base ` A )
22 mapdpglem26.t 
 |-  .x. = ( .s ` C )
23 mapdpglem26.o 
 |-  O = ( 0g ` A )
24 mapdpglem28.ve 
 |-  ( ph -> v e. B )
25 mapdpglem28.u1 
 |-  ( ph -> h = ( u .x. i ) )
26 mapdpglem28.u2 
 |-  ( ph -> ( G R h ) = ( v .x. ( G R i ) ) )
27 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
28 1 3 12 dvhlmod 
 |-  ( ph -> U e. LMod )
29 13 eldifad 
 |-  ( ph -> X e. V )
30 4 27 7 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
31 28 29 30 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
32 14 eldifad 
 |-  ( ph -> Y e. V )
33 4 27 7 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
34 28 32 33 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
35 1 3 27 2 12 31 34 mapd11 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) = ( M ` ( N ` { Y } ) ) <-> ( N ` { X } ) = ( N ` { Y } ) ) )
36 35 necon3bid 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) =/= ( M ` ( N ` { Y } ) ) <-> ( N ` { X } ) =/= ( N ` { Y } ) ) )
37 16 36 mpbird 
 |-  ( ph -> ( M ` ( N ` { X } ) ) =/= ( M ` ( N ` { Y } ) ) )
38 19 simprd 
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) )
39 38 simpld 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { i } ) )
40 37 17 39 3netr3d 
 |-  ( ph -> ( J ` { G } ) =/= ( J ` { i } ) )