Metamath Proof Explorer

Theorem mapdpglem31

Description: Lemma for mapdpg . Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-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 ) ) )
mapdpglem28.ue 
|- ( ph -> u e. B )
Assertion mapdpglem31 
|- ( ph -> h = 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 mapdpglem28.ue 
 |-  ( ph -> u e. B )
28 eqid 
 |-  ( 1r ` A ) = ( 1r ` A )
29 eqid 
 |-  ( Scalar ` C ) = ( Scalar ` C )
30 eqid 
 |-  ( 1r ` ( Scalar ` C ) ) = ( 1r ` ( Scalar ` C ) )
31 1 3 20 28 8 29 30 12 lcd1 
 |-  ( ph -> ( 1r ` ( Scalar ` C ) ) = ( 1r ` A ) )
32 31 oveq1d 
 |-  ( ph -> ( ( 1r ` ( Scalar ` C ) ) .x. i ) = ( ( 1r ` A ) .x. i ) )
33 1 8 12 lcdlmod 
 |-  ( ph -> C e. LMod )
34 19 simpld 
 |-  ( ph -> i e. F )
35 9 29 22 30 lmodvs1 
 |-  ( ( C e. LMod /\ i e. F ) -> ( ( 1r ` ( Scalar ` C ) ) .x. i ) = i )
36 33 34 35 syl2anc 
 |-  ( ph -> ( ( 1r ` ( Scalar ` C ) ) .x. i ) = i )
37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 mapdpglem30 
 |-  ( ph -> ( v = ( 1r ` A ) /\ v = u ) )
38 eqtr2 
 |-  ( ( v = ( 1r ` A ) /\ v = u ) -> ( 1r ` A ) = u )
39 37 38 syl 
 |-  ( ph -> ( 1r ` A ) = u )
40 39 oveq1d 
 |-  ( ph -> ( ( 1r ` A ) .x. i ) = ( u .x. i ) )
41 32 36 40 3eqtr3rd 
 |-  ( ph -> ( u .x. i ) = i )
42 25 41 eqtrd 
 |-  ( ph -> h = i )