Metamath Proof Explorer

Theorem hdmap1l6h

Description: Lemmma for hdmap1l6 . Part (6) of Baer p. 48 line 2. (Contributed by NM, 1-May-2015)

Ref Expression
Hypotheses hdmap1l6.h 
|- H = ( LHyp ` K )
hdmap1l6.u 
|- U = ( ( DVecH ` K ) ` W )
hdmap1l6.v 
|- V = ( Base ` U )
hdmap1l6.p 
|- .+ = ( +g ` U )
hdmap1l6.s 
|- .- = ( -g ` U )
hdmap1l6c.o 
|- .0. = ( 0g ` U )
hdmap1l6.n 
|- N = ( LSpan ` U )
hdmap1l6.c 
|- C = ( ( LCDual ` K ) ` W )
hdmap1l6.d 
|- D = ( Base ` C )
hdmap1l6.a 
|- .+b = ( +g ` C )
hdmap1l6.r 
|- R = ( -g ` C )
hdmap1l6.q 
|- Q = ( 0g ` C )
hdmap1l6.l 
|- L = ( LSpan ` C )
hdmap1l6.m 
|- M = ( ( mapd ` K ) ` W )
hdmap1l6.i 
|- I = ( ( HDMap1 ` K ) ` W )
hdmap1l6.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmap1l6.f 
|- ( ph -> F e. D )
hdmap1l6cl.x 
|- ( ph -> X e. ( V \ { .0. } ) )
hdmap1l6.mn 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
hdmap1l6d.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
hdmap1l6d.yz 
|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
hdmap1l6d.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
hdmap1l6d.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
hdmap1l6d.w 
|- ( ph -> w e. ( V \ { .0. } ) )
hdmap1l6d.wn 
|- ( ph -> -. w e. ( N ` { X , Y } ) )
Assertion hdmap1l6h 
|- ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )

Proof

Step Hyp Ref Expression
1 hdmap1l6.h 
 |-  H = ( LHyp ` K )
2 hdmap1l6.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmap1l6.v 
 |-  V = ( Base ` U )
4 hdmap1l6.p 
 |-  .+ = ( +g ` U )
5 hdmap1l6.s 
 |-  .- = ( -g ` U )
6 hdmap1l6c.o 
 |-  .0. = ( 0g ` U )
7 hdmap1l6.n 
 |-  N = ( LSpan ` U )
8 hdmap1l6.c 
 |-  C = ( ( LCDual ` K ) ` W )
9 hdmap1l6.d 
 |-  D = ( Base ` C )
10 hdmap1l6.a 
 |-  .+b = ( +g ` C )
11 hdmap1l6.r 
 |-  R = ( -g ` C )
12 hdmap1l6.q 
 |-  Q = ( 0g ` C )
13 hdmap1l6.l 
 |-  L = ( LSpan ` C )
14 hdmap1l6.m 
 |-  M = ( ( mapd ` K ) ` W )
15 hdmap1l6.i 
 |-  I = ( ( HDMap1 ` K ) ` W )
16 hdmap1l6.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
17 hdmap1l6.f 
 |-  ( ph -> F e. D )
18 hdmap1l6cl.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
19 hdmap1l6.mn 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( L ` { F } ) )
20 hdmap1l6d.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
21 hdmap1l6d.yz 
 |-  ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
22 hdmap1l6d.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
23 hdmap1l6d.z 
 |-  ( ph -> Z e. ( V \ { .0. } ) )
24 hdmap1l6d.w 
 |-  ( ph -> w e. ( V \ { .0. } ) )
25 hdmap1l6d.wn 
 |-  ( ph -> -. w e. ( N ` { X , Y } ) )
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 hdmap1l6g 
 |-  ( ph -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , Y >. ) ) .+b ( I ` <. X , F , Z >. ) ) )
27 1 8 16 lcdlmod 
 |-  ( ph -> C e. LMod )
28 1 2 16 dvhlvec 
 |-  ( ph -> U e. LVec )
29 24 eldifad 
 |-  ( ph -> w e. V )
30 18 eldifad 
 |-  ( ph -> X e. V )
31 22 eldifad 
 |-  ( ph -> Y e. V )
32 3 7 28 29 30 31 25 lspindpi 
 |-  ( ph -> ( ( N ` { w } ) =/= ( N ` { X } ) /\ ( N ` { w } ) =/= ( N ` { Y } ) ) )
33 32 simpld 
 |-  ( ph -> ( N ` { w } ) =/= ( N ` { X } ) )
34 33 necomd 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { w } ) )
35 1 2 3 6 7 8 9 13 14 15 16 17 19 34 18 29 hdmap1cl 
 |-  ( ph -> ( I ` <. X , F , w >. ) e. D )
36 23 eldifad 
 |-  ( ph -> Z e. V )
37 3 7 28 30 31 36 20 lspindpi 
 |-  ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) )
38 37 simpld 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
39 1 2 3 6 7 8 9 13 14 15 16 17 19 38 18 31 hdmap1cl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
40 37 simprd 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
41 1 2 3 6 7 8 9 13 14 15 16 17 19 40 18 36 hdmap1cl 
 |-  ( ph -> ( I ` <. X , F , Z >. ) e. D )
42 9 10 lmodass 
 |-  ( ( C e. LMod /\ ( ( I ` <. X , F , w >. ) e. D /\ ( I ` <. X , F , Y >. ) e. D /\ ( I ` <. X , F , Z >. ) e. D ) ) -> ( ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , Y >. ) ) .+b ( I ` <. X , F , Z >. ) ) = ( ( I ` <. X , F , w >. ) .+b ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) )
43 27 35 39 41 42 syl13anc 
 |-  ( ph -> ( ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , Y >. ) ) .+b ( I ` <. X , F , Z >. ) ) = ( ( I ` <. X , F , w >. ) .+b ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) )
44 26 43 eqtrd 
 |-  ( ph -> ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) )
45 3 4 6 7 28 18 22 23 24 21 38 25 mapdindp1 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) )
46 1 2 16 dvhlmod 
 |-  ( ph -> U e. LMod )
47 3 4 lmodvacl 
 |-  ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
48 46 31 36 47 syl3anc 
 |-  ( ph -> ( Y .+ Z ) e. V )
49 1 2 3 6 7 8 9 13 14 15 16 17 19 45 18 48 hdmap1cl 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) e. D )
50 9 10 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 )
51 27 39 41 50 syl3anc 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) e. D )
52 9 10 lmodlcan 
 |-  ( ( C e. LMod /\ ( ( I ` <. X , F , ( Y .+ Z ) >. ) e. D /\ ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) e. D /\ ( I ` <. X , F , w >. ) e. D ) ) -> ( ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) <-> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) )
53 27 49 51 35 52 syl13anc 
 |-  ( ph -> ( ( ( I ` <. X , F , w >. ) .+b ( I ` <. X , F , ( Y .+ Z ) >. ) ) = ( ( I ` <. X , F , w >. ) .+b ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) <-> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) ) )
54 44 53 mpbid 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) = ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) )