Theorem lspfval 14020

Description: The span function for a left vector space (or 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
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		df-lsp 14019	. . 3 |- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
3		fveq2 5561	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lspval.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2247	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3611	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5561	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		lspval.s	. . . . . . 7 |- S = ( LSubSp ` W )
9	7, 8	eqtr4di 2247	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = S )
10	9	rabeqdv 2757	. . . . 5 |- ( w = W -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
11	10	inteqd 3880	. . . 4 |- ( w = W -> |^| { t e. ( LSubSp ` w ) | s C_ t } = |^| { t e. S | s C_ t } )
12	6, 11	mpteq12dv 4116	. . 3 |- ( w = W -> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
13		elex 2774	. . 3 |- ( W e. X -> W e. _V )
14		basfn 12761	. . . . . . 7 |- Base Fn _V
15		funfvex 5578	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5361	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
17	14, 13, 16	sylancr 414	. . . . . 6 |- ( W e. X -> ( Base ` W ) e. _V )
18	4, 17	eqeltrid 2283	. . . . 5 |- ( W e. X -> V e. _V )
19	18	pwexd 4215	. . . 4 |- ( W e. X -> ~P V e. _V )
20	19	mptexd 5792	. . 3 |- ( W e. X -> ( s e. ~P V |-> |^| { t e. S | s C_ t } ) e. _V )
21	2, 12, 13, 20	fvmptd3 5658	. 2 |- ( W e. X -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
22	1, 21	eqtrid 2241	1 |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2167   {crab 2479   _Vcvv 2763   C_ wss 3157   ~Pcpw 3606   |^|cint 3875   |-> cmpt 4095   Fn wfn 5254   ` cfv 5259   Basecbs 12703   LSubSpclss 13984   LSpanclspn 14018

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-lsp 14019

This theorem is referenced by:  lspf  14021  lspval  14022  lspex  14027  lsppropd  14064