Metamath Proof Explorer

Theorem mapdh6lem2N

Description: Lemma for mapdh6N . Part (6) in Baer p. 47, lines 20-22. (Contributed by NM, 13-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 )
mapdhe6.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdhe6.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
mapdhe6.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
mapdh6.yz 
|- ( ph -> ( N ` { Y } ) =/= ( N ` { Z } ) )
mapdh6.fg 
|- ( ph -> ( I ` <. X , F , Y >. ) = G )
mapdh6.fe 
|- ( ph -> ( I ` <. X , F , Z >. ) = E )
Assertion mapdh6lem2N 
|- ( ph -> ( M ` ( N ` { ( Y .+ Z ) } ) ) = ( J ` { ( G .+b E ) } ) )

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 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 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
27 3 5 14 dvhlmod 
 |-  ( ph -> U e. LMod )
28 20 eldifad 
 |-  ( ph -> Y e. V )
29 6 26 9 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
30 27 28 29 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
31 21 eldifad 
 |-  ( ph -> Z e. V )
32 6 26 9 lspsncl 
 |-  ( ( U e. LMod /\ Z e. V ) -> ( N ` { Z } ) e. ( LSubSp ` U ) )
33 27 31 32 syl2anc 
 |-  ( ph -> ( N ` { Z } ) e. ( LSubSp ` U ) )
34 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
35 26 34 lsmcl 
 |-  ( ( U e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` U ) /\ ( N ` { Z } ) e. ( LSubSp ` U ) ) -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) e. ( LSubSp ` U ) )
36 27 30 33 35 syl3anc 
 |-  ( ph -> ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) e. ( LSubSp ` U ) )
37 17 eldifad 
 |-  ( ph -> X e. V )
38 6 18 lmodvacl 
 |-  ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
39 27 28 31 38 syl3anc 
 |-  ( ph -> ( Y .+ Z ) e. V )
40 6 7 lmodvsubcl 
 |-  ( ( U e. LMod /\ X e. V /\ ( Y .+ Z ) e. V ) -> ( X .- ( Y .+ Z ) ) e. V )
41 27 37 39 40 syl3anc 
 |-  ( ph -> ( X .- ( Y .+ Z ) ) e. V )
42 6 26 9 lspsncl 
 |-  ( ( U e. LMod /\ ( X .- ( Y .+ Z ) ) e. V ) -> ( N ` { ( X .- ( Y .+ Z ) ) } ) e. ( LSubSp ` U ) )
43 27 41 42 syl2anc 
 |-  ( ph -> ( N ` { ( X .- ( Y .+ Z ) ) } ) e. ( LSubSp ` U ) )
44 6 26 9 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
45 27 37 44 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
46 26 34 lsmcl 
 |-  ( ( U e. LMod /\ ( N ` { ( X .- ( Y .+ Z ) ) } ) e. ( LSubSp ` U ) /\ ( N ` { X } ) e. ( LSubSp ` U ) ) -> ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) e. ( LSubSp ` U ) )
47 27 43 45 46 syl3anc 
 |-  ( ph -> ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) e. ( LSubSp ` U ) )
48 3 4 5 26 14 36 47 mapdin 
 |-  ( ph -> ( M ` ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) ) = ( ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) i^i ( M ` ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) ) )
49 eqid 
 |-  ( LSSum ` C ) = ( LSSum ` C )
50 3 4 5 26 34 10 49 14 30 33 mapdlsm 
 |-  ( ph -> ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) = ( ( M ` ( N ` { Y } ) ) ( LSSum ` C ) ( M ` ( N ` { Z } ) ) ) )
51 3 5 14 dvhlvec 
 |-  ( ph -> U e. LVec )
52 6 8 9 51 28 21 37 23 22 lspindp2 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ -. Z e. ( N ` { X , Y } ) ) )
53 52 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 53 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
55 24 54 eqeltrrd 
 |-  ( ph -> G e. D )
56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20 55 53 mapdheq 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) = G <-> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) ) )
57 24 56 mpbid 
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) = ( J ` { G } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( F R G ) } ) ) )
58 57 simpld 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
59 6 8 9 51 20 31 37 23 22 lspindp1 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Z } ) /\ -. Y e. ( N ` { X , Z } ) ) )
60 59 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
61 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 31 60 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Z >. ) e. D )
62 25 61 eqeltrrd 
 |-  ( ph -> E e. D )
63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 21 62 60 mapdheq 
 |-  ( ph -> ( ( I ` <. X , F , Z >. ) = E <-> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) ) ) )
64 25 63 mpbid 
 |-  ( ph -> ( ( M ` ( N ` { Z } ) ) = ( J ` { E } ) /\ ( M ` ( N ` { ( X .- Z ) } ) ) = ( J ` { ( F R E ) } ) ) )
65 64 simpld 
 |-  ( ph -> ( M ` ( N ` { Z } ) ) = ( J ` { E } ) )
66 58 65 oveq12d 
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) ( LSSum ` C ) ( M ` ( N ` { Z } ) ) ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
67 50 66 eqtrd 
 |-  ( ph -> ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
68 3 4 5 26 34 10 49 14 43 45 mapdlsm 
 |-  ( ph -> ( M ` ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) = ( ( M ` ( N ` { ( X .- ( Y .+ Z ) ) } ) ) ( LSSum ` C ) ( M ` ( N ` { X } ) ) ) )
69 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 ) ) } ) )
70 69 16 oveq12d 
 |-  ( ph -> ( ( M ` ( N ` { ( X .- ( Y .+ Z ) ) } ) ) ( LSSum ` C ) ( M ` ( N ` { X } ) ) ) = ( ( J ` { ( F R ( G .+b E ) ) } ) ( LSSum ` C ) ( J ` { F } ) ) )
71 68 70 eqtrd 
 |-  ( ph -> ( M ` ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) = ( ( J ` { ( F R ( G .+b E ) ) } ) ( LSSum ` C ) ( J ` { F } ) ) )
72 67 71 ineq12d 
 |-  ( ph -> ( ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) i^i ( M ` ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) ) = ( ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) i^i ( ( J ` { ( F R ( G .+b E ) ) } ) ( LSSum ` C ) ( J ` { F } ) ) ) )
73 48 72 eqtrd 
 |-  ( ph -> ( M ` ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) ) = ( ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) i^i ( ( J ` { ( F R ( G .+b E ) ) } ) ( LSSum ` C ) ( J ` { F } ) ) ) )
74 6 7 8 34 9 51 37 22 23 20 21 18 baerlem5b 
 |-  ( ph -> ( N ` { ( Y .+ Z ) } ) = ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) )
75 74 fveq2d 
 |-  ( ph -> ( M ` ( N ` { ( Y .+ Z ) } ) ) = ( M ` ( ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) i^i ( ( N ` { ( X .- ( Y .+ Z ) ) } ) ( LSSum ` U ) ( N ` { X } ) ) ) ) )
76 3 10 14 lcdlvec 
 |-  ( ph -> C e. LVec )
77 3 4 5 6 9 10 11 13 14 15 16 37 28 55 58 31 62 65 22 mapdindp 
 |-  ( ph -> -. F e. ( J ` { G , E } ) )
78 3 4 5 6 9 10 11 13 14 55 58 28 31 62 65 23 mapdncol 
 |-  ( ph -> ( J ` { G } ) =/= ( J ` { E } ) )
79 3 4 5 6 9 10 11 13 14 55 58 8 1 20 mapdn0 
 |-  ( ph -> G e. ( D \ { Q } ) )
80 3 4 5 6 9 10 11 13 14 62 65 8 1 21 mapdn0 
 |-  ( ph -> E e. ( D \ { Q } ) )
81 11 12 1 49 13 76 15 77 78 79 80 19 baerlem5b 
 |-  ( ph -> ( J ` { ( G .+b E ) } ) = ( ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) i^i ( ( J ` { ( F R ( G .+b E ) ) } ) ( LSSum ` C ) ( J ` { F } ) ) ) )
82 73 75 81 3eqtr4d 
 |-  ( ph -> ( M ` ( N ` { ( Y .+ Z ) } ) ) = ( J ` { ( G .+b E ) } ) )