Metamath Proof Explorer


Theorem hdmaplem2N

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) (New usage is discouraged.)

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 )
Assertion hdmaplem2N
|- ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )

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 elun2
 |-  ( Z e. ( N ` { Y } ) -> Z e. ( ( N ` { X } ) u. ( N ` { Y } ) ) )
8 5 7 nsyl
 |-  ( ph -> -. Z e. ( N ` { Y } ) )
9 1 2 3 4 6 8 lspsnne2
 |-  ( ph -> ( N ` { Z } ) =/= ( N ` { Y } ) )