Theorem lspfval 14835

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 14834	. . 3 |- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
3		fveq2 5698	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lspval.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2289	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3693	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5698	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		lspval.s	. . . . . . 7 |- S = ( LSubSp ` W )
9	7, 8	eqtr4di 2289	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = S )
10	9	rabeqdv 2815	. . . . 5 |- ( w = W -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
11	10	inteqd 3975	. . . 4 |- ( w = W -> |^| { t e. ( LSubSp ` w ) | s C_ t } = |^| { t e. S | s C_ t } )
12	6, 11	mpteq12dv 4214	. . 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 2833	. . 3 |- ( W e. X -> W e. _V )
14		basfn 13470	. . . . . . 7 |- Base Fn _V
15		funfvex 5715	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5486	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
17	14, 13, 16	sylancr 418	. . . . . 6 |- ( W e. X -> ( Base ` W ) e. _V )
18	4, 17	eqeltrid 2325	. . . . 5 |- ( W e. X -> V e. _V )
19	18	pwexd 4320	. . . 4 |- ( W e. X -> ~P V e. _V )
20	19	mptexd 5947	. . 3 |- ( W e. X -> ( s e. ~P V |-> |^| { t e. S | s C_ t } ) e. _V )
21	2, 12, 13, 20	fvmptd3 5802	. 2 |- ( W e. X -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
22	1, 21	eqtrid 2283	1 |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   |^|cint 3970   |-> cmpt 4193   Fn wfn 5375   ` cfv 5380   Basecbs 13411   LSubSpclss 14799   LSpanclspn 14833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8277  ax-resscn 8278  ax-1re 8280  ax-addrcl 8283

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-inn 9315  df-ndx 13414  df-slot 13415  df-base 13417  df-lsp 14834

This theorem is used by:  lspf  14836  lspval  14837  lspex  14842  lsppropd  14879