Metamath Proof Explorer

Theorem mapdh6cN

Description: Lemmma for mapdh6N . (Contributed by NM, 24-Apr-2015) (New usage is discouraged.)

Ref Expression
Hypotheses mapdh.q 
|- Q = ( 0g ` C )
mapdh.i 
|- I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
mapdh.h 
|- H = ( LHyp ` K )
mapdh.m 
|- M = ( ( mapd ` K ) ` W )
mapdh.u 
|- U = ( ( DVecH ` K ) ` W )
mapdh.v 
|- V = ( Base ` U )
mapdh.s 
|- .- = ( -g ` U )
mapdhc.o 
|- .0. = ( 0g ` U )
mapdh.n 
|- N = ( LSpan ` U )
mapdh.c 
|- C = ( ( LCDual ` K ) ` W )
mapdh.d 
|- D = ( Base ` C )
mapdh.r 
|- R = ( -g ` C )
mapdh.j 
|- J = ( LSpan ` C )
mapdh.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdhc.f 
|- ( ph -> F e. D )
mapdh.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdhcl.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdh.p 
|- .+ = ( +g ` U )
mapdh.a 
|- .+b = ( +g ` C )
mapdh6c.y 
|- ( ph -> Y e. V )
mapdh6c.z 
|- ( ph -> Z = .0. )
mapdh6c.ne 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion mapdh6cN 
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )

Proof

Step Hyp Ref Expression
1 mapdh.q 
 |-  Q = ( 0g ` C )
2 mapdh.i 
 |-  I = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
3 mapdh.h 
 |-  H = ( LHyp ` K )
4 mapdh.m 
 |-  M = ( ( mapd ` K ) ` W )
5 mapdh.u 
 |-  U = ( ( DVecH ` K ) ` W )
6 mapdh.v 
 |-  V = ( Base ` U )
7 mapdh.s 
 |-  .- = ( -g ` U )
8 mapdhc.o 
 |-  .0. = ( 0g ` U )
9 mapdh.n 
 |-  N = ( LSpan ` U )
10 mapdh.c 
 |-  C = ( ( LCDual ` K ) ` W )
11 mapdh.d 
 |-  D = ( Base ` C )
12 mapdh.r 
 |-  R = ( -g ` C )
13 mapdh.j 
 |-  J = ( LSpan ` C )
14 mapdh.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
15 mapdhc.f 
 |-  ( ph -> F e. D )
16 mapdh.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
17 mapdhcl.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
18 mapdh.p 
 |-  .+ = ( +g ` U )
19 mapdh.a 
 |-  .+b = ( +g ` C )
20 mapdh6c.y 
 |-  ( ph -> Y e. V )
21 mapdh6c.z 
 |-  ( ph -> Z = .0. )
22 mapdh6c.ne 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
23 3 10 14 lcdlmod 
 |-  ( ph -> C e. LMod )
24 lmodgrp 
 |-  ( C e. LMod -> C e. Grp )
25 23 24 syl 
 |-  ( ph -> C e. Grp )
26 3 5 14 dvhlvec 
 |-  ( ph -> U e. LVec )
27 17 eldifad 
 |-  ( ph -> X e. V )
28 3 5 14 dvhlmod 
 |-  ( ph -> U e. LMod )
29 6 8 lmod0vcl 
 |-  ( U e. LMod -> .0. e. V )
30 28 29 syl 
 |-  ( ph -> .0. e. V )
31 21 30 eqeltrd 
 |-  ( ph -> Z e. V )
32 6 9 26 27 20 31 22 lspindpi 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
33 32 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 33 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
35 11 19 1 grprid 
 |-  ( ( C e. Grp /\ ( I ` <. X , F , Y >. ) e. D ) -> ( ( I ` <. X , F , Y >. ) .+b Q ) = ( I ` <. X , F , Y >. ) )
36 25 34 35 syl2anc 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) .+b Q ) = ( I ` <. X , F , Y >. ) )
37 21 oteq3d 
 |-  ( ph -> <. X , F , Z >. = <. X , F , .0. >. )
38 37 fveq2d 
 |-  ( ph -> ( I ` <. X , F , Z >. ) = ( I ` <. X , F , .0. >. ) )
39 1 2 8 17 15 mapdhval0 
 |-  ( ph -> ( I ` <. X , F , .0. >. ) = Q )
40 38 39 eqtrd 
 |-  ( ph -> ( I ` <. X , F , Z >. ) = Q )
41 40 oveq2d 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) = ( ( I ` <. X , F , Y >. ) .+b Q ) )
42 21 oveq2d 
 |-  ( ph -> ( Y .+ Z ) = ( Y .+ .0. ) )
43 lmodgrp 
 |-  ( U e. LMod -> U e. Grp )
44 28 43 syl 
 |-  ( ph -> U e. Grp )
45 6 18 8 grprid 
 |-  ( ( U e. Grp /\ Y e. V ) -> ( Y .+ .0. ) = Y )
46 44 20 45 syl2anc 
 |-  ( ph -> ( Y .+ .0. ) = Y )
47 42 46 eqtrd 
 |-  ( ph -> ( Y .+ Z ) = Y )
48 47 oteq3d 
 |-  ( ph -> <. X , F , ( Y .+ Z ) >. = <. X , F , Y >. )
49 48 fveq2d 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , Y >. ) )
50 36 41 49 3eqtr4rd 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )