Theorem lspfval 14337

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 14336	. . 3 |- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
3		fveq2 5623	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lspval.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2280	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3654	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5623	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		lspval.s	. . . . . . 7 |- S = ( LSubSp ` W )
9	7, 8	eqtr4di 2280	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = S )
10	9	rabeqdv 2793	. . . . 5 |- ( w = W -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
11	10	inteqd 3927	. . . 4 |- ( w = W -> |^| { t e. ( LSubSp ` w ) | s C_ t } = |^| { t e. S | s C_ t } )
12	6, 11	mpteq12dv 4165	. . 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 2811	. . 3 |- ( W e. X -> W e. _V )
14		basfn 13077	. . . . . . 7 |- Base Fn _V
15		funfvex 5640	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5419	. . . . . . 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 2316	. . . . 5 |- ( W e. X -> V e. _V )
19	18	pwexd 4264	. . . 4 |- ( W e. X -> ~P V e. _V )
20	19	mptexd 5859	. . 3 |- ( W e. X -> ( s e. ~P V |-> |^| { t e. S | s C_ t } ) e. _V )
21	2, 12, 13, 20	fvmptd3 5721	. 2 |- ( W e. X -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
22	1, 21	eqtrid 2274	1 |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   |^|cint 3922   |-> cmpt 4144   Fn wfn 5309   ` cfv 5314   Basecbs 13018   LSubSpclss 14301   LSpanclspn 14335

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 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

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-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-inn 9099  df-ndx 13021  df-slot 13022  df-base 13024  df-lsp 14336

This theorem is referenced by:  lspf  14338  lspval  14339  lspex  14344  lsppropd  14381