Metamath Proof Explorer

Theorem mapdpglem27

Description: Lemma for mapdpg . Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (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 )
Assertion mapdpglem27 
|- ( ph -> E. v e. ( B \ { O } ) ( G R h ) = ( v .x. ( G R 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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 mapdpglem25 
 |-  ( ph -> ( ( J ` { h } ) = ( J ` { i } ) /\ ( J ` { ( G R h ) } ) = ( J ` { ( G R i ) } ) ) )
25 24 simprd 
 |-  ( ph -> ( J ` { ( G R h ) } ) = ( J ` { ( G R i ) } ) )
26 eqid 
 |-  ( Scalar ` C ) = ( Scalar ` C )
27 eqid 
 |-  ( Base ` ( Scalar ` C ) ) = ( Base ` ( Scalar ` C ) )
28 eqid 
 |-  ( 0g ` ( Scalar ` C ) ) = ( 0g ` ( Scalar ` C ) )
29 1 8 12 lcdlvec 
 |-  ( ph -> C e. LVec )
30 1 8 12 lcdlmod 
 |-  ( ph -> C e. LMod )
31 18 simpld 
 |-  ( ph -> h e. F )
32 9 10 lmodvsubcl 
 |-  ( ( C e. LMod /\ G e. F /\ h e. F ) -> ( G R h ) e. F )
33 30 15 31 32 syl3anc 
 |-  ( ph -> ( G R h ) e. F )
34 19 simpld 
 |-  ( ph -> i e. F )
35 9 10 lmodvsubcl 
 |-  ( ( C e. LMod /\ G e. F /\ i e. F ) -> ( G R i ) e. F )
36 30 15 34 35 syl3anc 
 |-  ( ph -> ( G R i ) e. F )
37 9 26 27 28 22 11 29 33 36 lspsneq 
 |-  ( ph -> ( ( J ` { ( G R h ) } ) = ( J ` { ( G R i ) } ) <-> E. v e. ( ( Base ` ( Scalar ` C ) ) \ { ( 0g ` ( Scalar ` C ) ) } ) ( G R h ) = ( v .x. ( G R i ) ) ) )
38 1 3 20 21 8 26 27 12 lcdsbase 
 |-  ( ph -> ( Base ` ( Scalar ` C ) ) = B )
39 1 3 20 23 8 26 28 12 lcd0 
 |-  ( ph -> ( 0g ` ( Scalar ` C ) ) = O )
40 39 sneqd 
 |-  ( ph -> { ( 0g ` ( Scalar ` C ) ) } = { O } )
41 38 40 difeq12d 
 |-  ( ph -> ( ( Base ` ( Scalar ` C ) ) \ { ( 0g ` ( Scalar ` C ) ) } ) = ( B \ { O } ) )
42 41 rexeqdv 
 |-  ( ph -> ( E. v e. ( ( Base ` ( Scalar ` C ) ) \ { ( 0g ` ( Scalar ` C ) ) } ) ( G R h ) = ( v .x. ( G R i ) ) <-> E. v e. ( B \ { O } ) ( G R h ) = ( v .x. ( G R i ) ) ) )
43 37 42 bitrd 
 |-  ( ph -> ( ( J ` { ( G R h ) } ) = ( J ` { ( G R i ) } ) <-> E. v e. ( B \ { O } ) ( G R h ) = ( v .x. ( G R i ) ) ) )
44 25 43 mpbid 
 |-  ( ph -> E. v e. ( B \ { O } ) ( G R h ) = ( v .x. ( G R i ) ) )