Metamath Proof Explorer

Theorem lspss

Description: Span preserves subset ordering. ( spanss analog.) (Contributed by NM, 11-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lspss.v 
|- V = ( Base ` W )
lspss.n 
|- N = ( LSpan ` W )
Assertion lspss 
|- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )

Proof

Step Hyp Ref Expression
1 lspss.v 
 |-  V = ( Base ` W )
2 lspss.n 
 |-  N = ( LSpan ` W )
3 simpl3 
 |-  ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> T C_ U )
4 sstr2 
 |-  ( T C_ U -> ( U C_ t -> T C_ t ) )
5 3 4 syl 
 |-  ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> ( U C_ t -> T C_ t ) )
6 5 ss2rabdv 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> { t e. ( LSubSp ` W ) | U C_ t } C_ { t e. ( LSubSp ` W ) | T C_ t } )
7 intss 
 |-  ( { t e. ( LSubSp ` W ) | U C_ t } C_ { t e. ( LSubSp ` W ) | T C_ t } -> |^| { t e. ( LSubSp ` W ) | T C_ t } C_ |^| { t e. ( LSubSp ` W ) | U C_ t } )
8 6 7 syl 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> |^| { t e. ( LSubSp ` W ) | T C_ t } C_ |^| { t e. ( LSubSp ` W ) | U C_ t } )
9 simp1 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> W e. LMod )
10 simp3 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ U )
11 simp2 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> U C_ V )
12 10 11 sstrd 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ V )
13 eqid 
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
14 1 13 2 lspval 
 |-  ( ( W e. LMod /\ T C_ V ) -> ( N ` T ) = |^| { t e. ( LSubSp ` W ) | T C_ t } )
15 9 12 14 syl2anc 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) = |^| { t e. ( LSubSp ` W ) | T C_ t } )
16 1 13 2 lspval 
 |-  ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
17 16 3adant3 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
18 8 15 17 3sstr4d 
 |-  ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )