Theorem dihsmsprn

Description: Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015)

	Ref	Expression
Hypotheses	dihsmsprn.h	|- H = ( LHyp ` K )
	dihsmsprn.u	|- U = ( ( DVecH ` K ) ` W )
	dihsmsprn.v	|- V = ( Base ` U )
	dihsmsprn.p	|- .(+) = ( LSSum ` U )
	dihsmsprn.n	|- N = ( LSpan ` U )
	dihsmsprn.i	|- I = ( ( DIsoH ` K ) ` W )
	dihsmsprn.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	dihsmsprn.x	|- ( ph -> X e. ran I )
	dihsmsprn.t	|- ( ph -> T e. V )
Assertion	dihsmsprn	|- ( ph -> ( X .(+) ( N ` { T } ) ) e. ran I )

Proof

Step	Hyp	Ref	Expression
1		dihsmsprn.h	|- H = ( LHyp ` K )
2		dihsmsprn.u	|- U = ( ( DVecH ` K ) ` W )
3		dihsmsprn.v	|- V = ( Base ` U )
4		dihsmsprn.p	|- .(+) = ( LSSum ` U )
5		dihsmsprn.n	|- N = ( LSpan ` U )
6		dihsmsprn.i	|- I = ( ( DIsoH ` K ) ` W )
7		dihsmsprn.k	|- ( ph -> ( K e. HL /\ W e. H ) )
8		dihsmsprn.x	|- ( ph -> X e. ran I )
9		dihsmsprn.t	|- ( ph -> T e. V )
10		eqid	|- ( ( joinH ` K ) ` W ) = ( ( joinH ` K ) ` W )
11	1 2 3 4 5 6 10 7 8 9	dihjat1	|- ( ph -> ( X ( ( joinH ` K ) ` W ) ( N ` { T } ) ) = ( X .(+) ( N ` { T } ) ) )
12	1 2 6 3	dihrnss	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X C_ V )
13	7 8 12	syl2anc	|- ( ph -> X C_ V )
14	1 2 7	dvhlmod	|- ( ph -> U e. LMod )
15	9	snssd	|- ( ph -> { T } C_ V )
16	3 5	lspssv	|- ( ( U e. LMod /\ { T } C_ V ) -> ( N ` { T } ) C_ V )
17	14 15 16	syl2anc	|- ( ph -> ( N ` { T } ) C_ V )
18	1 6 2 3 10	djhcl	|- ( ( ( K e. HL /\ W e. H ) /\ ( X C_ V /\ ( N ` { T } ) C_ V ) ) -> ( X ( ( joinH ` K ) ` W ) ( N ` { T } ) ) e. ran I )
19	7 13 17 18	syl12anc	|- ( ph -> ( X ( ( joinH ` K ) ` W ) ( N ` { T } ) ) e. ran I )
20	11 19	eqeltrrd	|- ( ph -> ( X .(+) ( N ` { T } ) ) e. ran I )