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Metamath Proof Explorer

Theorem lsmsp2

Description: Subspace sum of spans of subsets is the span of their union. ( spanuni analog.) (Contributed by NM, 22-Feb-2014) (Revised by Mario Carneiro, 21-Jun-2014)

Ref Expression
Hypotheses lsmsp2.v 
|- V = ( Base ` W )
lsmsp2.n 
|- N = ( LSpan ` W )
lsmsp2.p 
|- .(+) = ( LSSum ` W )
Assertion lsmsp2 
|- ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( ( N ` T ) .(+) ( N ` U ) ) = ( N ` ( T u. U ) ) )

Proof

Step Hyp Ref Expression
1 lsmsp2.v 
 |-  V = ( Base ` W )
2 lsmsp2.n 
 |-  N = ( LSpan ` W )
3 lsmsp2.p 
 |-  .(+) = ( LSSum ` W )
4 simp1 
 |-  ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> W e. LMod )
5 eqid 
 |-  ( LSubSp ` W ) = ( LSubSp ` W )
6 1 5 2 lspcl 
 |-  ( ( W e. LMod /\ T C_ V ) -> ( N ` T ) e. ( LSubSp ` W ) )
7 6 3adant3 
 |-  ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( N ` T ) e. ( LSubSp ` W ) )
8 1 5 2 lspcl 
 |-  ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) e. ( LSubSp ` W ) )
9 8 3adant2 
 |-  ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( N ` U ) e. ( LSubSp ` W ) )
10 5 2 3 lsmsp 
 |-  ( ( W e. LMod /\ ( N ` T ) e. ( LSubSp ` W ) /\ ( N ` U ) e. ( LSubSp ` W ) ) -> ( ( N ` T ) .(+) ( N ` U ) ) = ( N ` ( ( N ` T ) u. ( N ` U ) ) ) )
11 4 7 9 10 syl3anc 
 |-  ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( ( N ` T ) .(+) ( N ` U ) ) = ( N ` ( ( N ` T ) u. ( N ` U ) ) ) )
12 1 2 lspun 
 |-  ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( N ` ( T u. U ) ) = ( N ` ( ( N ` T ) u. ( N ` U ) ) ) )
13 11 12 eqtr4d 
 |-  ( ( W e. LMod /\ T C_ V /\ U C_ V ) -> ( ( N ` T ) .(+) ( N ` U ) ) = ( N ` ( T u. U ) ) )