Theorem lspval 14727

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 14725	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 5697	. . 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 2238	. . 3 |- ( s e. ~P V |-> |^| { t e. S | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } )
8		sseq1 3271	. . . . 5 |- ( s = U -> ( s C_ t <-> U C_ t ) )
9	8	rabbidv 2810	. . . 4 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
10	9	inteqd 3975	. . 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 13411	. . . . . . 7 |- Base Fn _V
13		elex 2833	. . . . . . . 8 |- ( W e. LMod -> W e. _V )
14	13	adantr 276	. . . . . . 7 |- ( ( W e. LMod /\ U C_ V ) -> W e. _V )
15		funfvex 5712	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5483	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
17	12, 14, 16	sylancr 418	. . . . . 6 |- ( ( W e. LMod /\ U C_ V ) -> ( Base ` W ) e. _V )
18	1, 17	eqeltrid 2325	. . . . 5 |- ( ( W e. LMod /\ U C_ V ) -> V e. _V )
19		elpw2g 4292	. . . . 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 14699	. . . . 5 |- ( W e. LMod -> V e. S )
23		sseq2 3272	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
24	23	rspcev 2929	. . . . 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 4289	. . . 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 5799	. 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 2271	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   |^|cint 3970   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377   Basecbs 13352   LModclmod 14623   LSubSpclss 14689   LSpanclspn 14723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-mulr 13445  df-sca 13447  df-vsca 13448  df-0g 13612  df-mgm 13676  df-sgrp 13717  df-mnd 13730  df-grp 13808  df-lmod 14625  df-lssm 14690  df-lsp 14724

This theorem is used by:  lspid  14734  lspss  14736  lspssid  14737