Theorem lspval 14538

Description: The span of a set of vectors (in a left module). (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 14536	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 5672	. . 3 |- ( W e. LMod -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
6	5	adantr 276	. 2 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
7		eqid 2232	. . 3 |- ( s e. ~P V |-> |^| { t e. S | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } )
8		sseq1 3261	. . . . 5 |- ( s = U -> ( s C_ t <-> U C_ t ) )
9	8	rabbidv 2802	. . . 4 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
10	9	inteqd 3954	. . 3 |- ( s = U -> |^| { t e. S | s C_ t } = |^| { t e. S | U C_ t } )
11		simpr 110	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> U C_ V )
12		basfn 13271	. . . . . . 7 |- Base Fn _V
13		elex 2825	. . . . . . . 8 |- ( W e. LMod -> W e. _V )
14	13	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ U C_ V ) -> W e. _V )
15		funfvex 5687	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5458	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
17	12, 14, 16	sylancr 414	. . . . . 6 |- ( ( W e. LMod /\ U C_ V ) -> ( Base ` W ) e. _V )
18	1, 17	eqeltrid 2319	. . . . 5 |- ( ( W e. LMod /\ U C_ V ) -> V e. _V )
19		elpw2g 4268	. . . . 5 |- ( V e. _V -> ( U e. ~P V <-> U C_ V ) )
20	18, 19	syl 14	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> ( U e. ~P V <-> U C_ V ) )
21	11, 20	mpbird 167	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> U e. ~P V )
22	1, 2	lss1 14510	. . . . 5 |- ( W e. LMod -> V e. S )
23		sseq2 3262	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
24	23	rspcev 2921	. . . . 5 |- ( ( V e. S /\ U C_ V ) -> E. t e. S U C_ t )
25	22, 24	sylan 283	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> E. t e. S U C_ t )
26		intexrabim 4265	. . . 4 |- ( E. t e. S U C_ t -> |^| { t e. S | U C_ t } e. _V )
27	25, 26	syl 14	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> |^| { t e. S | U C_ t } e. _V )
28	7, 10, 21, 27	fvmptd3 5771	. 2 |- ( ( W e. LMod /\ U C_ V ) -> ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) = |^| { t e. S | U C_ t } )
29	6, 28	eqtrd 2265	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E.wrex 2521   {crab 2524   _Vcvv 2813   C_ wss 3211   ~Pcpw 3669   |^|cint 3949   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352   Basecbs 13212   LModclmod 14435   LSubSpclss 14500   LSpanclspn 14534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-lmod 14437  df-lssm 14501  df-lsp 14535

This theorem is referenced by:  lspid  14545  lspss  14547  lspssid  14548