Metamath Proof Explorer

Theorem lspabs2

Description: Absorption law for span of vector sum. (Contributed by NM, 30-Apr-2015)

Ref Expression
Hypotheses lspabs2.v 
|- V = ( Base ` W )
lspabs2.p 
|- .+ = ( +g ` W )
lspabs2.o 
|- .0. = ( 0g ` W )
lspabs2.n 
|- N = ( LSpan ` W )
lspabs2.w 
|- ( ph -> W e. LVec )
lspabs2.x 
|- ( ph -> X e. V )
lspabs2.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
lspabs2.e 
|- ( ph -> ( N ` { X } ) = ( N ` { ( X .+ Y ) } ) )
Assertion lspabs2 
|- ( ph -> ( N ` { X } ) = ( N ` { Y } ) )

Proof

Step Hyp Ref Expression
1 lspabs2.v 
 |-  V = ( Base ` W )
2 lspabs2.p 
 |-  .+ = ( +g ` W )
3 lspabs2.o 
 |-  .0. = ( 0g ` W )
4 lspabs2.n 
 |-  N = ( LSpan ` W )
5 lspabs2.w 
 |-  ( ph -> W e. LVec )
6 lspabs2.x 
 |-  ( ph -> X e. V )
7 lspabs2.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
8 lspabs2.e 
 |-  ( ph -> ( N ` { X } ) = ( N ` { ( X .+ Y ) } ) )
9 lveclmod 
 |-  ( W e. LVec -> W e. LMod )
10 5 9 syl 
 |-  ( ph -> W e. LMod )
11 1 4 lspsnsubg 
 |-  ( ( W e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( SubGrp ` W ) )
12 10 6 11 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( SubGrp ` W ) )
13 7 eldifad 
 |-  ( ph -> Y e. V )
14 1 4 lspsnsubg 
 |-  ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( SubGrp ` W ) )
15 10 13 14 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( SubGrp ` W ) )
16 eqid 
 |-  ( LSSum ` W ) = ( LSSum ` W )
17 16 lsmub2 
 |-  ( ( ( N ` { X } ) e. ( SubGrp ` W ) /\ ( N ` { Y } ) e. ( SubGrp ` W ) ) -> ( N ` { Y } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
18 12 15 17 syl2anc 
 |-  ( ph -> ( N ` { Y } ) C_ ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
19 8 oveq2d 
 |-  ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { X } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
20 16 lsmidm 
 |-  ( ( N ` { X } ) e. ( SubGrp ` W ) -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { X } ) ) = ( N ` { X } ) )
21 12 20 syl 
 |-  ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { X } ) ) = ( N ` { X } ) )
22 1 2 4 10 6 13 lspprabs 
 |-  ( ph -> ( N ` { X , ( X .+ Y ) } ) = ( N ` { X , Y } ) )
23 1 2 lmodvacl 
 |-  ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( X .+ Y ) e. V )
24 10 6 13 23 syl3anc 
 |-  ( ph -> ( X .+ Y ) e. V )
25 1 4 16 10 6 24 lsmpr 
 |-  ( ph -> ( N ` { X , ( X .+ Y ) } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) )
26 1 4 16 10 6 13 lsmpr 
 |-  ( ph -> ( N ` { X , Y } ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
27 22 25 26 3eqtr3d 
 |-  ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { ( X .+ Y ) } ) ) = ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) )
28 19 21 27 3eqtr3rd 
 |-  ( ph -> ( ( N ` { X } ) ( LSSum ` W ) ( N ` { Y } ) ) = ( N ` { X } ) )
29 18 28 sseqtrd 
 |-  ( ph -> ( N ` { Y } ) C_ ( N ` { X } ) )
30 1 3 4 5 7 6 lspsncmp 
 |-  ( ph -> ( ( N ` { Y } ) C_ ( N ` { X } ) <-> ( N ` { Y } ) = ( N ` { X } ) ) )
31 29 30 mpbid 
 |-  ( ph -> ( N ` { Y } ) = ( N ` { X } ) )
32 31 eqcomd 
 |-  ( ph -> ( N ` { X } ) = ( N ` { Y } ) )