| 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 >. ) ) ) |