Metamath Proof Explorer

Theorem mapdh6hN

Description: Lemmma for mapdh6N . Part (6) of Baer p. 48 line 2. (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 mapdh6hN 
|- ( 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 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 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 mapdh6gN 
 |-  ( 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 3 10 14 lcdlmod 
 |-  ( ph -> C e. LMod )
28 24 eldifad 
 |-  ( ph -> w e. V )
29 3 5 14 dvhlvec 
 |-  ( ph -> U e. LVec )
30 17 eldifad 
 |-  ( ph -> X e. V )
31 22 eldifad 
 |-  ( ph -> Y e. V )
32 6 9 29 28 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 4 5 6 7 8 9 10 11 12 13 14 15 16 17 28 34 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , w >. ) e. D )
36 23 eldifad 
 |-  ( ph -> Z e. V )
37 6 9 29 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 4 5 6 7 8 9 10 11 12 13 14 15 16 17 31 38 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Y >. ) e. D )
40 37 simprd 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
41 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 36 40 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , Z >. ) e. D )
42 11 19 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 5 14 dvhlmod 
 |-  ( ph -> U e. LMod )
46 6 18 lmodvacl 
 |-  ( ( U e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
47 45 31 36 46 syl3anc 
 |-  ( ph -> ( Y .+ Z ) e. V )
48 6 18 8 9 29 17 22 23 24 21 38 25 mapdindp1 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { ( Y .+ Z ) } ) )
49 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 47 48 mapdhcl 
 |-  ( ph -> ( I ` <. X , F , ( Y .+ Z ) >. ) e. D )
50 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 )
51 27 39 41 50 syl3anc 
 |-  ( ph -> ( ( I ` <. X , F , Y >. ) .+b ( I ` <. X , F , Z >. ) ) e. D )
52 11 19 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 >. ) ) )