Theorem lspval 18975

Description: The span of a set of vectors (in a left module). (spanval 28192 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 18973	. . . 4 |- ( W e. LMod -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
5	4	fveq1d 6193	. . 3 |- ( W e. LMod -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
6	5	adantr 481	. 2 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = ( ( s e. ~P V |-> |^| { t e. S | s C_ t } ) ` U ) )
7		simpr 477	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> U C_ V )
8		fvex 6201	. . . . . 6 |- ( Base ` W ) e. _V
9	1, 8	eqeltri 2697	. . . . 5 |- V e. _V
10	9	elpw2 4828	. . . 4 |- ( U e. ~P V <-> U C_ V )
11	7, 10	sylibr 224	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> U e. ~P V )
12	1, 2	lss1 18939	. . . . 5 |- ( W e. LMod -> V e. S )
13		sseq2 3627	. . . . . 6 |- ( t = V -> ( U C_ t <-> U C_ V ) )
14	13	rspcev 3309	. . . . 5 |- ( ( V e. S /\ U C_ V ) -> E. t e. S U C_ t )
15	12, 14	sylan 488	. . . 4 |- ( ( W e. LMod /\ U C_ V ) -> E. t e. S U C_ t )
16		intexrab 4823	. . . 4 |- ( E. t e. S U C_ t <-> |^| { t e. S | U C_ t } e. _V )
17	15, 16	sylib 208	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> |^| { t e. S | U C_ t } e. _V )
18		sseq1 3626	. . . . . 6 |- ( s = U -> ( s C_ t <-> U C_ t ) )
19	18	rabbidv 3189	. . . . 5 |- ( s = U -> { t e. S | s C_ t } = { t e. S | U C_ t } )
20	19	inteqd 4480	. . . 4 |- ( s = U -> |^| { t e. S | s C_ t } = |^| { t e. S | U C_ t } )
21		eqid 2622	. . . 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 6280	. . 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 693	. 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 2656	1 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. S | U C_ t } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-lmod 18865  df-lss 18933  df-lsp 18972

This theorem is referenced by:  lspid  18982  lspss  18984  lspssid  18985  dochspss  36667  lcosslsp  42227