| 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 |
|
mapdh6d.xn |
|- ( ph -> -. X e. ( N ` { Y , Z } ) ) |
| 21 |
|
mapdh6d.yz |
|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) ) |
| 22 |
|
mapdh6d.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
| 23 |
|
mapdh6d.z |
|- ( ph -> Z e. ( V \ { .0. } ) ) |
| 24 |
|
mapdh6d.w |
|- ( ph -> w e. ( V \ { .0. } ) ) |
| 25 |
|
mapdh6d.wn |
|- ( ph -> -. w e. ( N ` { X , Y } ) ) |
| 26 |
3 5 14
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 27 |
24
|
eldifad |
|- ( ph -> w e. V ) |
| 28 |
22
|
eldifad |
|- ( ph -> Y e. V ) |
| 29 |
6 18
|
lmodvacl |
|- ( ( U e. LMod /\ w e. V /\ Y e. V ) -> ( w .+ Y ) e. V ) |
| 30 |
26 27 28 29
|
syl3anc |
|- ( ph -> ( w .+ Y ) e. V ) |
| 31 |
3 5 14
|
dvhlvec |
|- ( ph -> U e. LVec ) |
| 32 |
17
|
eldifad |
|- ( ph -> X e. V ) |
| 33 |
6 9 31 27 32 28 25
|
lspindpi |
|- ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) ) |
| 34 |
33
|
simprd |
|- ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) ) |
| 35 |
6 18 8 9 26 27 28 34
|
lmodindp1 |
|- ( ph -> ( w .+ Y ) =/= .0. ) |
| 36 |
|
eldifsn |
|- ( ( w .+ Y ) e. ( V \ { .0. } ) <-> ( ( w .+ Y ) e. V /\ ( w .+ Y ) =/= .0. ) ) |
| 37 |
30 35 36
|
sylanbrc |
|- ( ph -> ( w .+ Y ) e. ( V \ { .0. } ) ) |
| 38 |
23
|
eldifad |
|- ( ph -> Z e. V ) |
| 39 |
6 9 31 32 28 38 20
|
lspindpi |
|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) ) |
| 40 |
39
|
simpld |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
| 41 |
6 18 8 9 31 17 22 23 24 21 40 25
|
mapdindp3 |
|- ( ph -> ( N ` { X } ) =/= ( N ` { ( w .+ Y ) } ) ) |
| 42 |
6 18 8 9 31 17 22 23 24 21 40 25
|
mapdindp4 |
|- ( ph -> -. Z e. ( N ` { X , ( w .+ Y ) } ) ) |
| 43 |
6 8 9 31 17 30 38 41 42
|
lspindp1 |
|- ( ph -> ( ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) /\ -. X e. ( N ` { Z , ( w .+ Y ) } ) ) ) |
| 44 |
43
|
simprd |
|- ( ph -> -. X e. ( N ` { Z , ( w .+ Y ) } ) ) |
| 45 |
|
prcom |
|- { ( w .+ Y ) , Z } = { Z , ( w .+ Y ) } |
| 46 |
45
|
fveq2i |
|- ( N ` { ( w .+ Y ) , Z } ) = ( N ` { Z , ( w .+ Y ) } ) |
| 47 |
46
|
eleq2i |
|- ( X e. ( N ` { ( w .+ Y ) , Z } ) <-> X e. ( N ` { Z , ( w .+ Y ) } ) ) |
| 48 |
44 47
|
sylnibr |
|- ( ph -> -. X e. ( N ` { ( w .+ Y ) , Z } ) ) |
| 49 |
6 9 31 38 32 30 42
|
lspindpi |
|- ( ph -> ( ( N ` { Z } ) =/= ( N ` { X } ) /\ ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) ) ) |
| 50 |
49
|
simprd |
|- ( ph -> ( N ` { Z } ) =/= ( N ` { ( w .+ Y ) } ) ) |
| 51 |
50
|
necomd |
|- ( ph -> ( N ` { ( w .+ Y ) } ) =/= ( N ` { Z } ) ) |
| 52 |
|
eqidd |
|- ( ph -> ( I ` <. X , F , ( w .+ Y ) >. ) = ( I ` <. X , F , ( w .+ Y ) >. ) ) |
| 53 |
|
eqidd |
|- ( ph -> ( I ` <. X , F , Z >. ) = ( I ` <. X , F , Z >. ) ) |
| 54 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 37 23 48 51 52 53
|
mapdh6aN |
|- ( ph -> ( I ` <. X , F , ( ( w .+ Y ) .+ Z ) >. ) = ( ( I ` <. X , F , ( w .+ Y ) >. ) .+b ( I ` <. X , F , Z >. ) ) ) |