Metamath Proof Explorer

Theorem aspss

Description: Span preserves subset ordering. ( spanss analog.) (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Hypotheses aspval.a 
|- A = ( AlgSpan ` W )
aspval.v 
|- V = ( Base ` W )
Assertion aspss 
|- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` T ) C_ ( A ` S ) )

Proof

Step Hyp Ref Expression
1 aspval.a 
 |-  A = ( AlgSpan ` W )
2 aspval.v 
 |-  V = ( Base ` W )
3 simpl3 
 |-  ( ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) /\ t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) ) -> T C_ S )
4 sstr2 
 |-  ( T C_ S -> ( S C_ t -> T C_ t ) )
5 3 4 syl 
 |-  ( ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) /\ t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) ) -> ( S C_ t -> T C_ t ) )
6 5 ss2rabdv 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } C_ { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } )
7 intss 
 |-  ( { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } C_ { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } -> |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } C_ |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
8 6 7 syl 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } C_ |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
9 simp1 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> W e. AssAlg )
10 simp3 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> T C_ S )
11 simp2 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> S C_ V )
12 10 11 sstrd 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> T C_ V )
13 eqid 
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
14 1 2 13 aspval 
 |-  ( ( W e. AssAlg /\ T C_ V ) -> ( A ` T ) = |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } )
15 9 12 14 syl2anc 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` T ) = |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } )
16 1 2 13 aspval 
 |-  ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
17 16 3adant3 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` S ) = |^| { t e. ( ( SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
18 8 15 17 3sstr4d 
 |-  ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` T ) C_ ( A ` S ) )