Metamath Proof Explorer

Theorem hdmap1l6b

Description: Lemmma for hdmap1l6 . (Contributed by NM, 24-Apr-2015)

Ref Expression
Hypotheses hdmap1l6.h 
|- H = ( LHyp ` K )
hdmap1l6.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap1l6.v 
|- V = ( Base ` U )
hdmap1l6.p 
|- .+ = ( +g ` U )
hdmap1l6.s 
|- .- = ( -g ` U )
hdmap1l6c.o 
|- .0. = ( 0g ` U )
hdmap1l6.n 
|- N = ( LSpan ` U )
hdmap1l6.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap1l6.d 
|- D = ( Base ` C )
hdmap1l6.a 
|- .+b = ( +g ` C )
hdmap1l6.r 
|- R = ( -g ` C )
hdmap1l6.q 
|- Q = ( 0g ` C )
hdmap1l6.l 
|- L = ( LSpan ` C )
hdmap1l6.m 
|- M = ( ( mapd ` K ) ` W )
hdmap1l6.i 
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1l6.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1l6.f 
|- ( ph -> F e. D )
hdmap1l6cl.x 
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1l6.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1l6b.y 
|- ( ph -> Y = .0. )
hdmap1l6b.z 
|- ( ph -> Z e. V )
hdmap1l6b.ne 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion hdmap1l6b 
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1l6.h 
 |-  H = ( LHyp ` K )
2 hdmap1l6.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1l6.v 
 |-  V = ( Base ` U )
4 hdmap1l6.p 
 |-  .+ = ( +g ` U )
5 hdmap1l6.s 
 |-  .- = ( -g ` U )
6 hdmap1l6c.o 
 |-  .0. = ( 0g ` U )
7 hdmap1l6.n 
 |-  N = ( LSpan ` U )
8 hdmap1l6.c 
 |-  C = ( ( LCDual ` K ) ` W )
9 hdmap1l6.d 
 |-  D = ( Base ` C )
10 hdmap1l6.a 
 |-  .+b = ( +g ` C )
11 hdmap1l6.r 
 |-  R = ( -g ` C )
12 hdmap1l6.q 
 |-  Q = ( 0g ` C )
13 hdmap1l6.l 
 |-  L = ( LSpan ` C )
14 hdmap1l6.m 
 |-  M = ( ( mapd ` K ) ` W )
15 hdmap1l6.i 
 |-  I = ( ( HDMap1 ` K ) ` W )
16 hdmap1l6.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
17 hdmap1l6.f 
 |-  ( ph -> F e. D )
18 hdmap1l6cl.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
19 hdmap1l6.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
20 hdmap1l6b.y 
 |-  ( ph -> Y = .0. )
21 hdmap1l6b.z 
 |-  ( ph -> Z e. V )
22 hdmap1l6b.ne 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
23 1 8 16 lcdlmod 
 |-  ( ph -> C e. LMod )
24 lmodgrp 
 |-  ( C e. LMod -> C e. Grp )
25 23 24 syl 
 |-  ( ph -> C e. Grp )
26 1 2 16 dvhlvec 
 |-  ( ph -> U e. LVec )
27 18 eldifad 
 |-  ( ph -> X e. V )
28 1 2 16 dvhlmod 
 |-  ( ph -> U e. LMod )
29 3 6 lmod0vcl 
 |-  ( U e. LMod -> .0. e. V )
30 28 29 syl 
 |-  ( ph -> .0. e. V )
31 20 30 eqeltrd 
 |-  ( ph -> Y e. V )
32 3 7 26 27 31 21 22 lspindpi 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
33 32 simprd 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
34 1 2 3 6 7 8 9 13 14 15 16 17 19 33 18 21 hdmap1cl 
 |-  ( ph -> ( I ` <. X , F , Z >. ) e. D )
35 9 10 12 grplid 
 |-  ( ( C e. Grp /\ ( I ` <. X , F , Z >. ) e. D ) -> ( Q .+b ( I ` <. X , F , Z >. ) ) = ( I ` <. X , F , Z >. ) )
36 25 34 35 syl2anc 
 |-  ( ph -> ( Q .+b ( I ` <. X , F , Z >. ) ) = ( I ` <. X , F , Z >. ) )
37 20 oteq3d 
 |-  ( ph -> <. X , F , Y >. = <. X , F , .0. >. )
38 37 fveq2d 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = ( I ` <. X , F , .0. >. ) )
39 1 2 3 6 8 9 12 15 16 17 27 hdmap1val0 
 |-  ( ph -> ( I ` <. X , F , .0. >. ) = Q )
40 38 39 eqtrd 
 |-  ( ph -> ( I ` <. X , F , Y >. ) = Q )
41 40 oveq1d 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) = ( Q .+b ( I ` <. X , F , Z >. ) ) )
42 20 oveq1d 
 |-  ( ph -> ( Y .+ Z ) = ( .0. .+ Z ) )
43 lmodgrp 
 |-  ( U e. LMod -> U e. Grp )
44 28 43 syl 
 |-  ( ph -> U e. Grp )
45 3 4 6 grplid 
 |-  ( ( U e. Grp /\ Z e. V ) -> ( .0. .+ Z ) = Z )
46 44 21 45 syl2anc 
 |-  ( ph -> ( .0. .+ Z ) = Z )
47 42 46 eqtrd 
 |-  ( ph -> ( Y .+ Z ) = Z )
48 47 oteq3d 
 |-  ( ph -> <. X , F , ( Y .+ Z ) >. = <. X , F , Z >. )
49 48 fveq2d 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , Z >. ) )
50 36 41 49 3eqtr4rd 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )