Metamath Proof Explorer

Theorem hdmaplem3

Description: Lemma to convert a frequently-used union condition. TODO: see if this can be applied to other hdmap* theorems. (Contributed by NM, 17-May-2015)

Ref Expression
Hypotheses hdmaplem1.v 
|- V = ( Base ` W )
hdmaplem1.n 
|- N = ( LSpan ` W )
hdmaplem1.w 
|- ( ph -> W e. LMod )
hdmaplem1.z 
|- ( ph -> Z e. V )
hdmaplem1.j 
|- ( ph -> -. Z e. ( ( N ` { X } ) u. ( N ` { Y } ) ) )
hdmaplem1.y 
|- ( ph -> Y e. V )
hdmaplem3.o 
|- .0. = ( 0g ` W )
Assertion hdmaplem3 
|- ( ph -> Z e. ( V \ { .0. } ) )

Proof

Step Hyp Ref Expression
1 hdmaplem1.v 
 |-  V = ( Base ` W )
2 hdmaplem1.n 
 |-  N = ( LSpan ` W )
3 hdmaplem1.w 
 |-  ( ph -> W e. LMod )
4 hdmaplem1.z 
 |-  ( ph -> Z e. V )
5 hdmaplem1.j 
 |-  ( ph -> -. Z e. ( ( N ` { X } ) u. ( N ` { Y } ) ) )
6 hdmaplem1.y 
 |-  ( ph -> Y e. V )
7 hdmaplem3.o 
 |-  .0. = ( 0g ` W )
8 eqid 
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
9 1 8 2 lspsncl 
 |-  ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
10 3 6 9 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
11 elun2 
 |-  ( Z e. ( N ` { Y } ) -> Z e. ( ( N ` { X } ) u. ( N ` { Y } ) ) )
12 5 11 nsyl 
 |-  ( ph -> -. Z e. ( N ` { Y } ) )
13 7 8 3 10 4 12 lssneln0 
 |-  ( ph -> Z e. ( V \ { .0. } ) )