Metamath Proof Explorer


Theorem mapdh6bN

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 )
mapdh6b.y
|- ( ph -> Y = .0. )
mapdh6b.z
|- ( ph -> Z e. V )
mapdh6b.ne
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion mapdh6bN
|- ( 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 mapdh6b.y
 |-  ( ph -> Y = .0. )
21 mapdh6b.z
 |-  ( ph -> Z e. V )
22 mapdh6b.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 20 30 eqeltrd
 |-  ( ph -> Y e. V )
32 6 9 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 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 33 mapdhcl
 |-  ( ph -> ( I ` <. X , F , Z >. ) e. D )
35 11 19 1 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 8 17 15 mapdhval0
 |-  ( 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 6 18 8 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 >. ) ) )