This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lspsnne1

Description: Two ways to express that vectors have different spans. (Contributed by NM, 28-May-2015)

Ref Expression
Hypotheses lspsnne1.v 
|- V = ( Base ` W )
lspsnne1.o 
|- .0. = ( 0g ` W )
lspsnne1.n 
|- N = ( LSpan ` W )
lspsnne1.w 
|- ( ph -> W e. LVec )
lspsnne1.x 
|- ( ph -> X e. ( V \ { .0. } ) )
lspsnne1.y 
|- ( ph -> Y e. V )
lspsnne1.e 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
Assertion lspsnne1 
|- ( ph -> -. X e. ( N ` { Y } ) )

Proof

Step Hyp Ref Expression
1 lspsnne1.v 
 |-  V = ( Base ` W )
2 lspsnne1.o 
 |-  .0. = ( 0g ` W )
3 lspsnne1.n 
 |-  N = ( LSpan ` W )
4 lspsnne1.w 
 |-  ( ph -> W e. LVec )
5 lspsnne1.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
6 lspsnne1.y 
 |-  ( ph -> Y e. V )
7 lspsnne1.e 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
8 eqid 
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
9 lveclmod 
 |-  ( W e. LVec -> W e. LMod )
10 4 9 syl 
 |-  ( ph -> W e. LMod )
11 1 8 3 lspsncl 
 |-  ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
12 10 6 11 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
13 5 eldifad 
 |-  ( ph -> X e. V )
14 1 8 3 10 12 13 ellspsn5b 
 |-  ( ph -> ( X e. ( N ` { Y } ) <-> ( N ` { X } ) C_ ( N ` { Y } ) ) )
15 14 notbid 
 |-  ( ph -> ( -. X e. ( N ` { Y } ) <-> -. ( N ` { X } ) C_ ( N ` { Y } ) ) )
16 1 2 3 4 5 6 lspsncmp 
 |-  ( ph -> ( ( N ` { X } ) C_ ( N ` { Y } ) <-> ( N ` { X } ) = ( N ` { Y } ) ) )
17 16 necon3bbid 
 |-  ( ph -> ( -. ( N ` { X } ) C_ ( N ` { Y } ) <-> ( N ` { X } ) =/= ( N ` { Y } ) ) )
18 15 17 bitrd 
 |-  ( ph -> ( -. X e. ( N ` { Y } ) <-> ( N ` { X } ) =/= ( N ` { Y } ) ) )
19 7 18 mpbird 
 |-  ( ph -> -. X e. ( N ` { Y } ) )