Theorem lspfval 18973

Description: The span function for a left vector space (or a left module). (df-span 28168 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
lspfval	|- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

Distinct variable groups:   t, s, S   V, s, t   W, s

Allowed substitution hints:   N( t, s)   W( t)   X( t, s)

Proof of Theorem lspfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspval.n	. 2 |- N = ( LSpan ` W )
2		elex 3212	. . 3 |- ( W e. X -> W e. _V )
3		fveq2 6191	. . . . . . 7 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lspval.v	. . . . . . 7 |- V = ( Base ` W )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 4163	. . . . 5 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		lspval.s	. . . . . . . 8 |- S = ( LSubSp ` W )
9	7, 8	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = S )
10		rabeq 3192	. . . . . . 7 |- ( ( LSubSp ` w ) = S -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
11	9, 10	syl 17	. . . . . 6 |- ( w = W -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
12	11	inteqd 4480	. . . . 5 |- ( w = W -> |^| { t e. ( LSubSp ` w ) | s C_ t } = |^| { t e. S | s C_ t } )
13	6, 12	mpteq12dv 4733	. . . 4 |- ( w = W -> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
14		df-lsp 18972	. . . 4 |- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
15		fvex 6201	. . . . . . 7 |- ( Base ` W ) e. _V
16	4, 15	eqeltri 2697	. . . . . 6 |- V e. _V
17	16	pwex 4848	. . . . 5 |- ~P V e. _V
18	17	mptex 6486	. . . 4 |- ( s e. ~P V |-> |^| { t e. S | s C_ t } ) e. _V
19	13, 14, 18	fvmpt 6282	. . 3 |- ( W e. _V -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
20	2, 19	syl 17	. 2 |- ( W e. X -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
21	1, 20	syl5eq 2668	1 |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   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-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-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-lsp 18972

This theorem is referenced by:  lspf  18974  lspval  18975  00lsp  18981  mrclsp  18989  lsppropd  19018