Metamath Proof Explorer

Theorem mapdh6dN

Description: Lemmma for mapdh6N . (Contributed by NM, 1-May-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 )
mapdh6d.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
mapdh6d.yz 
|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
mapdh6d.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdh6d.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
mapdh6d.w 
|- ( ph -> w e. ( V \ { .0. } ) )
mapdh6d.wn 
|- ( ph -> -. w e. ( N ` { X , Y } ) )
Assertion mapdh6dN 
|- ( ph -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ 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 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 10 14 lcdlmod 
 |-  ( ph -> C e. LMod )
27 24 eldifad 
 |-  ( ph -> w e. V )
28 3 5 14 dvhlvec 
 |-  ( ph -> U e. LVec )
29 17 eldifad 
 |-  ( ph -> X e. V )
30 22 eldifad 
 |-  ( ph -> Y e. V )
31 6 9 28 27 29 30 25 lspindpi 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
32 31 simpld 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { X } ) )
33 32 necomd 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { w } ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 27 33 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , w >. ) e. D )
35 11 19 1 lmod0vrid 
 |-  ( ( C e. LMod /\ ( I ` <. X , F , w >. ) e. D ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
36 26 34 35 syl2anc 
 |-  ( ph -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
37 36 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b Q ) = ( I ` <. X , F , w >. ) )
38 oteq3 
 |-  ( ( Y .+ Z ) = .0. -> <. X , F , ( Y .+ Z ) >. = <. X , F , .0. >. )
39 38 fveq2d 
 |-  ( ( Y .+ Z ) = .0. -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , .0. >. ) )
40 1 2 8 17 15 mapdhval0 
 |-  ( ph -> ( I ` <. X , F , .0. >. ) = Q )
41 39 40 sylan9eqr 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = Q )
42 41 oveq2d 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b Q ) )
43 oveq2 
 |-  ( ( Y .+ Z ) = .0. -> ( w .+ ( Y .+ Z ) ) = ( w .+ .0. ) )
44 3 5 14 dvhlmod 
 |-  ( ph -> U e. LMod )
45 6 18 8 lmod0vrid 
 |-  ( ( U e. LMod /\ w e. V ) -> ( w .+ .0. ) = w )
46 44 27 45 syl2anc 
 |-  ( ph -> ( w .+ .0. ) = w )
47 43 46 sylan9eqr 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( w .+ ( Y .+ Z ) ) = w )
48 47 oteq3d 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> <. X , F , ( w .+ ( Y .+ Z ) ) >. = <. X , F , w >. )
49 48 fveq2d 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( I ` <. X , F , w >. ) )
50 37 42 49 3eqtr4rd 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )
51 14 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( K e. HL /\ W e. H ) )
52 15 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> F e. D )
53 16 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
54 17 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> X e. ( V \ { .0. } ) )
55 24 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> w e. ( V \ { .0. } ) )
56 23 eldifad 
 |-  ( ph -> Z e. V )
57 6 18 lmodvacl 
 |-  ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
58 44 30 56 57 syl3anc 
 |-  ( ph -> ( Y .+ Z ) e. V )
59 58 anim1i 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
60 eldifsn 
 |-  ( ( Y .+ Z ) e. ( V \ { .0. } ) <-> ( ( Y .+ Z ) e. V /\ ( Y .+ Z ) =/= .0. ) )
61 59 60 sylibr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( Y .+ Z ) e. ( V \ { .0. } ) )
62 6 9 28 29 30 56 20 lspindpi 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
63 62 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
64 6 18 8 9 28 17 22 23 24 21 63 25 mapdindp1 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) )
65 6 18 8 9 28 17 22 23 24 21 63 25 mapdindp2 
 |-  ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )
66 6 8 9 28 17 58 27 64 65 lspindp1 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) /\ -. X e. ( N ` { w , ( Y .+ Z ) } ) ) )
67 66 simprd 
 |-  ( ph -> -. X e. ( N ` { w , ( Y .+ Z ) } ) )
68 67 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. X e. ( N ` { w , ( Y .+ Z ) } ) )
69 31 simprd 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { Y } ) )
70 6 8 9 28 24 30 69 lspsnne1 
 |-  ( ph -> -. w e. ( N ` { Y } ) )
71 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
72 6 9 71 44 30 56 lsmpr 
 |-  ( ph -> ( N ` { Y , Z } ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
73 21 oveq2d 
 |-  ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
74 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
75 6 74 9 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
76 44 30 75 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
77 74 lsssubg 
 |-  ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) ) -> ( N ` { Y } ) e. ( SubGrp ` U ) )
78 44 76 77 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( SubGrp ` U ) )
79 71 lsmidm 
 |-  ( ( N ` { Y } ) e. ( SubGrp ` U ) -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) )
80 78 79 syl 
 |-  ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Y } ) ) = ( N ` { Y } ) )
81 72 73 80 3eqtr2d 
 |-  ( ph -> ( N ` { Y , Z } ) = ( N ` { Y } ) )
82 70 81 neleqtrrd 
 |-  ( ph -> -. w e. ( N ` { Y , Z } ) )
83 6 18 9 44 30 56 27 82 lspindp4 
 |-  ( ph -> -. w e. ( N ` { Y , ( Y .+ Z ) } ) )
84 6 9 28 27 30 58 83 lspindpi 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { Y } ) /\ ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) ) )
85 84 simprd 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) )
86 85 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { w } ) =/= ( N ` { ( Y .+ Z ) } ) )
87 eqidd 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , w >. ) = ( I ` <. X , F , w >. ) )
88 eqidd 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( I ` <. X , F , ( Y .+ Z ) >. ) )
89 1 2 3 4 5 6 7 8 9 10 11 12 13 51 52 53 54 18 19 55 61 68 86 87 88 mapdh6aN 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )
90 50 89 pm2.61dane 
 |-  ( ph -> ( I ` <. X , F , ( w .+ ( Y .+ Z ) ) >. ) = ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) )