Theorem lspfval 13480

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 13479	. . 3 |- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
3		fveq2 5517	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lspval.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2228	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3582	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5517	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		lspval.s	. . . . . . 7 |- S = ( LSubSp ` W )
9	7, 8	eqtr4di 2228	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = S )
10	9	rabeqdv 2733	. . . . 5 |- ( w = W -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
11	10	inteqd 3851	. . . 4 |- ( w = W -> |^| { t e. ( LSubSp ` w ) | s C_ t } = |^| { t e. S | s C_ t } )
12	6, 11	mpteq12dv 4087	. . 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 2750	. . 3 |- ( W e. X -> W e. _V )
14		basfn 12522	. . . . . . 7 |- Base Fn _V
15		funfvex 5534	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5318	. . . . . . 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 2264	. . . . 5 |- ( W e. X -> V e. _V )
19	18	pwexd 4183	. . . 4 |- ( W e. X -> ~P V e. _V )
20	19	mptexd 5745	. . 3 |- ( W e. X -> ( s e. ~P V |-> |^| { t e. S | s C_ t } ) e. _V )
21	2, 12, 13, 20	fvmptd3 5611	. 2 |- ( W e. X -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
22	1, 21	eqtrid 2222	1 |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   {crab 2459   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   |^|cint 3846   |-> cmpt 4066   Fn wfn 5213   ` cfv 5218   Basecbs 12464   LSubSpclss 13447   LSpanclspn 13478

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-lsp 13479

This theorem is referenced by:  lspf  13481  lspval  13482  lsppropd  13523