Theorem lspval 16010

Description: The span of a set of vectors (in a left module). (spanval 22792 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lspval.v	|- V = ( Base ` W )
lspval.s	|- S = ( LSubSp ` W )
lspval.n	|- N = ( LSpan ` W )

Assertion

Ref	Expression
lspval	|- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Distinct variable groups:   t, S   t, U   t, V

Allowed substitution hints:   N( t)   W( t)

Proof of Theorem lspval

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspval.v	. . . . 5 |- V = ( Base ` W )
2		lspval.s	. . . . 5 |- S = ( LSubSp ` W )
3		lspval.n	. . . . 5 |- N = ( LSpan ` W )
4	1, 2, 3	lspfval 16008	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 5693	. . 3 |- ( W e. LMod -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
6	5	adantr 452	. 2 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
7		simpr 448	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> U C_ V )
8		fvex 5705	. . . . . 6 |- ( Base ` W ) e. _V
9	1, 8	eqeltri 2478	. . . . 5 |- V e. _V
10	9	elpw2 4328	. . . 4 |- ( U e. ~P V <-> U C_ V )
11	7, 10	sylibr 204	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> U e. ~P V )
12	1, 2	lss1 15974	. . . . 5 |- ( W e. LMod -> V e. S )
13		sseq2 3334	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
14	13	rspcev 3016	. . . . 5 |- ( ( V e. S /\ U C_ V ) -> E. t e. S U C_ t )
15	12, 14	sylan 458	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> E. t e. S U C_ t )
16		intexrab 4323	. . . 4 |- ( E. t e. S U C_ t <-> |^| { t e. S | U C_ t } e. _V )
17	15, 16	sylib 189	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> |^| { t e. S | U C_ t } e. _V )
18		sseq1 3333	. . . . . 6 |- ( s = U -> ( s C_ t <-> U C_ t ) )
19	18	rabbidv 2912	. . . . 5 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
20	19	inteqd 4019	. . . 4 |- ( s = U -> |^| { t e. S | s C_ t } = |^| { t e. S | U C_ t } )
21		eqid 2408	. . . 4 |- ( s e. ~P V |-> |^| { t e. S | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } )
22	20, 21	fvmptg 5767	. . 3 |- ( ( U e. ~P V /\ |^| { t e. S | U C_ t } e. _V ) -> ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) = |^| { t e. S | U C_ t } )
23	11, 17, 22	syl2anc 643	. 2 |- ( ( W e. LMod /\ U C_ V ) -> ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) = |^| { t e. S | U C_ t } )
24	6, 23	eqtrd 2440	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   E.wrex 2671   {crab 2674   _Vcvv 2920   C_ wss 3284   ~Pcpw 3763   |^|cint 4014   e. cmpt 4230   ` cfv 5417   Basecbs 13428   LModclmod 15909   LSubSpclss 15967   LSpanclspn 16006

This theorem is referenced by:  lspid  16017  lspss  16019  lspssid  16020  dochspss  31865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-riota 6512  df-0g 13686  df-mnd 14649  df-grp 14771  df-lmod 15911  df-lss 15968  df-lsp 16007