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Metamath Proof Explorer

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 )