Theorem lspval 15732

Description: The span of a set of vectors (in a left module). (spanval 21912 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 15730	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 5527	. . 3 |- ( W e. LMod -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
6	5	adantr 451	. 2 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
7		simpr 447	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> U C_ V )
8		fvex 5539	. . . . . 6 |- ( Base ` W ) e. _V
9	1, 8	eqeltri 2353	. . . . 5 |- V e. _V
10	9	elpw2 4175	. . . 4 |- ( U e. ~P V <-> U C_ V )
11	7, 10	sylibr 203	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> U e. ~P V )
12	1, 2	lss1 15696	. . . . 5 |- ( W e. LMod -> V e. S )
13		sseq2 3200	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
14	13	rspcev 2884	. . . . 5 |- ( ( V e. S /\ U C_ V ) -> E. t e. S U C_ t )
15	12, 14	sylan 457	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> E. t e. S U C_ t )
16		intexrab 4170	. . . 4 |- ( E. t e. S U C_ t <-> |^| { t e. S | U C_ t } e. _V )
17	15, 16	sylib 188	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> |^| { t e. S | U C_ t } e. _V )
18		sseq1 3199	. . . . . 6 |- ( s = U -> ( s C_ t <-> U C_ t ) )
19	18	rabbidv 2780	. . . . 5 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
20	19	inteqd 3867	. . . 4 |- ( s = U -> |^| { t e. S | s C_ t } = |^| { t e. S | U C_ t } )
21		eqid 2283	. . . 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 5600	. . 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 642	. 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 2315	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1623   e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788   C_ wss 3152   ~Pcpw 3625   |^|cint 3862   e. cmpt 4077   ` cfv 5255   Basecbs 13148   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728

This theorem is referenced by:  lspid  15739  lspss  15741  lspssid  15742  dochspss  31568

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-lmod 15629  df-lss 15690  df-lsp 15729