Metamath Proof Explorer

Theorem mapdh6aN

Description: Lemma for mapdh6N . Part (6) in Baer p. 47, case 1. (Contributed by NM, 23-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 mapdh6aN 
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , 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 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 ellspsn5 
 |-  ( ph -> ( N ` { ( Y .+ Z ) } ) C_ ( N ` { Y , Z } ) )
59 6 55 9 36 56 46 ellspsn5b 
 |-  ( 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 >. ) ) )