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 |
|
mapdhe6.y |
|- ( ph -> Y e. ( V \ { .0. } ) ) |
21 |
|
mapdhe6.z |
|- ( ph -> Z e. ( V \ { .0. } ) ) |
22 |
|
mapdhe6.xn |
|- ( ph -> -. X e. ( N ` { Y , Z } ) ) |
23 |
|
mapdh6.yz |
|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) ) |
24 |
|
mapdh6.fg |
|- ( ph -> ( I ` <. X , F , Y >. ) = G ) |
25 |
|
mapdh6.fe |
|- ( ph -> ( I ` <. X , F , Z >. ) = E ) |
26 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
|
mapdh6lem2N |
|- ( ph -> ( M ` ( N ` { ( Y .+ Z ) } ) ) = ( J ` { ( G .+b E ) } ) ) |
27 |
24 25
|
oveq12d |
|- ( ph -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) = ( G .+b E ) ) |
28 |
27
|
sneqd |
|- ( ph -> { ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) } = { ( G .+b E ) } ) |
29 |
28
|
fveq2d |
|- ( ph -> ( J ` { ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) } ) = ( J ` { ( G .+b E ) } ) ) |
30 |
26 29
|
eqtr4d |
|- ( ph -> ( M ` ( N ` { ( Y .+ Z ) } ) ) = ( J ` { ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) } ) ) |
31 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
|
mapdh6lem1N |
|- ( ph -> ( M ` ( N ` { ( X .- ( Y .+ Z ) ) } ) ) = ( J ` { ( F R ( G .+b E ) ) } ) ) |
32 |
27
|
oveq2d |
|- ( ph -> ( F R ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) = ( F R ( G .+b E ) ) ) |
33 |
32
|
sneqd |
|- ( ph -> { ( F R ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) } = { ( F R ( G .+b E ) ) } ) |
34 |
33
|
fveq2d |
|- ( ph -> ( J ` { ( F R ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) } ) = ( J ` { ( F R ( G .+b E ) ) } ) ) |
35 |
31 34
|
eqtr4d |
|- ( ph -> ( M ` ( N ` { ( X .- ( Y .+ Z ) ) } ) ) = ( J ` { ( F R ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) } ) ) |
36 |
3 5 14
|
dvhlmod |
|- ( ph -> U e. LMod ) |
37 |
20
|
eldifad |
|- ( ph -> Y e. V ) |
38 |
21
|
eldifad |
|- ( ph -> Z e. V ) |
39 |
6 18
|
lmodvacl |
|- ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V ) |
40 |
36 37 38 39
|
syl3anc |
|- ( ph -> ( Y .+ Z ) e. V ) |
41 |
6 18 8 9 36 37 38 23
|
lmodindp1 |
|- ( ph -> ( Y .+ Z ) =/= .0. ) |
42 |
|
eldifsn |
|- ( ( Y .+ Z ) e. ( V \ { .0. } ) <-> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) ) |
43 |
40 41 42
|
sylanbrc |
|- ( ph -> ( Y .+ Z ) e. ( V \ { .0. } ) ) |
44 |
3 10 14
|
lcdlmod |
|- ( ph -> C e. LMod ) |
45 |
3 5 14
|
dvhlvec |
|- ( ph -> U e. LVec ) |
46 |
17
|
eldifad |
|- ( ph -> X e. V ) |
47 |
6 8 9 45 37 21 46 23 22
|
lspindp2 |
|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ -. Z e. ( N ` { X , Y } ) ) ) |
48 |
47
|
simpld |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) ) |
49 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 37 48
|
mapdhcl |
|- ( ph -> ( I ` <. X , F , Y >. ) e. D ) |
50 |
6 8 9 45 20 38 46 23 22
|
lspindp1 |
|- ( ph -> ( ( N ` { X } ) =/= ( N ` { Z } ) /\ -. Y e. ( N ` { X , Z } ) ) ) |
51 |
50
|
simpld |
|- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) ) |
52 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 38 51
|
mapdhcl |
|- ( ph -> ( I ` <. X , F , Z >. ) e. D ) |
53 |
11 19
|
lmodvacl |
|- ( ( C e. LMod /\ ( I ` <. X , F , Y >. ) e. D /\ ( I ` <. X , F , Z >. ) e. D ) -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) e. D ) |
54 |
44 49 52 53
|
syl3anc |
|- ( ph -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) e. D ) |
55 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
56 |
6 55 9 36 37 38
|
lspprcl |
|- ( ph -> ( N ` { Y , Z } ) e. ( LSubSp ` U ) ) |
57 |
6 18 9 36 37 38
|
lspprvacl |
|- ( ph -> ( Y .+ Z ) e. ( N ` { Y , Z } ) ) |
58 |
55 9 36 56 57
|
lspsnel5a |
|- ( ph -> ( N ` { ( Y .+ Z ) } ) C_ ( N ` { Y , Z } ) ) |
59 |
6 55 9 36 56 46
|
lspsnel5 |
|- ( ph -> ( X e. ( N ` { Y , Z } ) <-> ( N ` { X } ) C_ ( N ` { Y , Z } ) ) ) |
60 |
22 59
|
mtbid |
|- ( ph -> -. ( N ` { X } ) C_ ( N ` { Y , Z } ) ) |
61 |
|
nssne2 |
|- ( ( ( N ` { ( Y .+ Z ) } ) C_ ( N ` { Y , Z } ) /\ -. ( N ` { X } ) C_ ( N ` { Y , Z } ) ) -> ( N ` { ( Y .+ Z ) } ) =/= ( N ` { X } ) ) |
62 |
58 60 61
|
syl2anc |
|- ( ph -> ( N ` { ( Y .+ Z ) } ) =/= ( N ` { X } ) ) |
63 |
62
|
necomd |
|- ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) ) |
64 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 43 54 63
|
mapdheq |
|- ( ph -> ( ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) <-> ( ( M ` ( N ` { ( Y .+ Z ) } ) ) = ( J ` { ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) } ) /\ ( M ` ( N ` { ( X .- ( Y .+ Z ) ) } ) ) = ( J ` { ( F R ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) } ) ) ) ) |
65 |
30 35 64
|
mpbir2and |
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) |