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 ) ) )