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 } ) )