Metamath Proof Explorer

Theorem mapdindp2

Description: Vector independence lemma. (Contributed by NM, 1-May-2015)

Ref Expression
Hypotheses mapdindp1.v 
|- V = ( Base ` W )
mapdindp1.p 
|- .+ = ( +g ` W )
mapdindp1.o 
|- .0. = ( 0g ` W )
mapdindp1.n 
|- N = ( LSpan ` W )
mapdindp1.w 
|- ( ph -> W e. LVec )
mapdindp1.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdindp1.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdindp1.z 
|- ( ph -> Z e. ( V \ { .0. } ) )
mapdindp1.W 
|- ( ph -> w e. ( V \ { .0. } ) )
mapdindp1.e 
|- ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
mapdindp1.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdindp1.f 
|- ( ph -> -. w e. ( N ` { X , Y } ) )
Assertion mapdindp2 
|- ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )

Proof

Step Hyp Ref Expression
1 mapdindp1.v 
 |-  V = ( Base ` W )
2 mapdindp1.p 
 |-  .+ = ( +g ` W )
3 mapdindp1.o 
 |-  .0. = ( 0g ` W )
4 mapdindp1.n 
 |-  N = ( LSpan ` W )
5 mapdindp1.w 
 |-  ( ph -> W e. LVec )
6 mapdindp1.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
7 mapdindp1.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
8 mapdindp1.z 
 |-  ( ph -> Z e. ( V \ { .0. } ) )
9 mapdindp1.W 
 |-  ( ph -> w e. ( V \ { .0. } ) )
10 mapdindp1.e 
 |-  ( ph -> ( N ` { Y } ) = ( N ` { Z } ) )
11 mapdindp1.ne 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
12 mapdindp1.f 
 |-  ( ph -> -. w e. ( N ` { X , Y } ) )
13 preq2 
 |-  ( ( Y .+ Z ) = .0. -> { X , ( Y .+ Z ) } = { X , .0. } )
14 13 fveq2d 
 |-  ( ( Y .+ Z ) = .0. -> ( N ` { X , ( Y .+ Z ) } ) = ( N ` { X , .0. } ) )
15 lveclmod 
 |-  ( W e. LVec -> W e. LMod )
16 5 15 syl 
 |-  ( ph -> W e. LMod )
17 6 eldifad 
 |-  ( ph -> X e. V )
18 1 3 4 16 17 lsppr0 
 |-  ( ph -> ( N ` { X , .0. } ) = ( N ` { X } ) )
19 14 18 sylan9eqr 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( N ` { X , ( Y .+ Z ) } ) = ( N ` { X } ) )
20 7 eldifad 
 |-  ( ph -> Y e. V )
21 prssi 
 |-  ( ( X e. V /\ Y e. V ) -> { X , Y } C_ V )
22 17 20 21 syl2anc 
 |-  ( ph -> { X , Y } C_ V )
23 snsspr1 
 |-  { X } C_ { X , Y }
24 23 a1i 
 |-  ( ph -> { X } C_ { X , Y } )
25 1 4 lspss 
 |-  ( ( W e. LMod /\ { X , Y } C_ V /\ { X } C_ { X , Y } ) -> ( N ` { X } ) C_ ( N ` { X , Y } ) )
26 16 22 24 25 syl3anc 
 |-  ( ph -> ( N ` { X } ) C_ ( N ` { X , Y } ) )
27 26 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( N ` { X } ) C_ ( N ` { X , Y } ) )
28 19 27 eqsstrd 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> ( N ` { X , ( Y .+ Z ) } ) C_ ( N ` { X , Y } ) )
29 12 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> -. w e. ( N ` { X , Y } ) )
30 28 29 ssneldd 
 |-  ( ( ph /\ ( Y .+ Z ) = .0. ) -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )
31 12 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. w e. ( N ` { X , Y } ) )
32 5 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> W e. LVec )
33 6 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> X e. ( V \ { .0. } ) )
34 7 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> Y e. ( V \ { .0. } ) )
35 8 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> Z e. ( V \ { .0. } ) )
36 9 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> w e. ( V \ { .0. } ) )
37 10 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { Y } ) = ( N ` { Z } ) )
38 11 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X } ) =/= ( N ` { Y } ) )
39 simpr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( Y .+ Z ) =/= .0. )
40 1 2 3 4 32 33 34 35 36 37 38 31 39 mapdindp0 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { ( Y .+ Z ) } ) = ( N ` { Y } ) )
41 40 oveq2d 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( Y .+ Z ) } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
42 eqid 
 |-  ( LSSum ` W ) = ( LSSum ` W )
43 8 eldifad 
 |-  ( ph -> Z e. V )
44 1 2 lmodvacl 
 |-  ( ( W e. LMod /\ Y e. V /\ Z e. V ) -> ( Y .+ Z ) e. V )
45 16 20 43 44 syl3anc 
 |-  ( ph -> ( Y .+ Z ) e. V )
46 1 4 42 16 17 45 lsmpr 
 |-  ( ph -> ( N ` { X , ( Y .+ Z ) } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( Y .+ Z ) } ) ) )
47 46 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X , ( Y .+ Z ) } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( Y .+ Z ) } ) ) )
48 1 4 42 16 17 20 lsmpr 
 |-  ( ph -> ( N ` { X , Y } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
49 48 adantr 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X , Y } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
50 41 47 49 3eqtr4d 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> ( N ` { X , ( Y .+ Z ) } ) = ( N ` { X , Y } ) )
51 31 50 neleqtrrd 
 |-  ( ( ph /\ ( Y .+ Z ) =/= .0. ) -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )
52 30 51 pm2.61dane 
 |-  ( ph -> -. w e. ( N ` { X , ( Y .+ Z ) } ) )