Theorem lspval 14394

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 14392	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 5637	. . 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 2229	. . 3 |- ( s e. ~P V |-> |^| { t e. S | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } )
8		sseq1 3248	. . . . 5 |- ( s = U -> ( s C_ t <-> U C_ t ) )
9	8	rabbidv 2789	. . . 4 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
10	9	inteqd 3931	. . 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 13131	. . . . . . 7 |- Base Fn _V
13		elex 2812	. . . . . . . 8 |- ( W e. LMod -> W e. _V )
14	13	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ U C_ V ) -> W e. _V )
15		funfvex 5652	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5429	. . . . . . 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 2316	. . . . 5 |- ( ( W e. LMod /\ U C_ V ) -> V e. _V )
19		elpw2g 4244	. . . . 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 14366	. . . . 5 |- ( W e. LMod -> V e. S )
23		sseq2 3249	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
24	23	rspcev 2908	. . . . 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 4241	. . . 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 5736	. 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 2262	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   _Vcvv 2800   C_ wss 3198   ~Pcpw 3650   |^|cint 3926   |-> cmpt 4148   Fn wfn 5319   ` cfv 5324   Basecbs 13072   LModclmod 14291   LSubSpclss 14356   LSpanclspn 14390

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-inn 9134  df-2 9192  df-3 9193  df-4 9194  df-5 9195  df-6 9196  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-mulr 13164  df-sca 13166  df-vsca 13167  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576  df-lmod 14293  df-lssm 14357  df-lsp 14391

This theorem is referenced by:  lspid  14401  lspss  14403  lspssid  14404