Theorem lspval 14403

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 14401	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 5641	. . 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 2231	. . 3 |- ( s e. ~P V |-> |^| { t e. S | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } )
8		sseq1 3250	. . . . 5 |- ( s = U -> ( s C_ t <-> U C_ t ) )
9	8	rabbidv 2791	. . . 4 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
10	9	inteqd 3933	. . 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 13140	. . . . . . 7 |- Base Fn _V
13		elex 2814	. . . . . . . 8 |- ( W e. LMod -> W e. _V )
14	13	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ U C_ V ) -> W e. _V )
15		funfvex 5656	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5432	. . . . . . 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 2318	. . . . 5 |- ( ( W e. LMod /\ U C_ V ) -> V e. _V )
19		elpw2g 4246	. . . . 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 14375	. . . . 5 |- ( W e. LMod -> V e. S )
23		sseq2 3251	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
24	23	rspcev 2910	. . . . 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 4243	. . . 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 5740	. 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 2264	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   {crab 2514   _Vcvv 2802   C_ wss 3200   ~Pcpw 3652   |^|cint 3928   |-> cmpt 4150   Fn wfn 5321   ` cfv 5326   Basecbs 13081   LModclmod 14300   LSubSpclss 14365   LSpanclspn 14399

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-lmod 14302  df-lssm 14366  df-lsp 14400

This theorem is referenced by:  lspid  14410  lspss  14412  lspssid  14413