Theorem lspfval 14220

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 14219	. . 3 |- LSpan = ( w e. _V |-> ( s e. ~P ( Base ` w ) |-> |^| { t e. ( LSubSp ` w ) | s C_ t } ) )
3		fveq2 5588	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lspval.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2257	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3625	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5588	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
8		lspval.s	. . . . . . 7 |- S = ( LSubSp ` W )
9	7, 8	eqtr4di 2257	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = S )
10	9	rabeqdv 2767	. . . . 5 |- ( w = W -> { t e. ( LSubSp ` w ) | s C_ t } = { t e. S | s C_ t } )
11	10	inteqd 3895	. . . 4 |- ( w = W -> |^| { t e. ( LSubSp ` w ) | s C_ t } = |^| { t e. S | s C_ t } )
12	6, 11	mpteq12dv 4133	. . 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 2785	. . 3 |- ( W e. X -> W e. _V )
14		basfn 12960	. . . . . . 7 |- Base Fn _V
15		funfvex 5605	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
16	15	funfni 5384	. . . . . . 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 2293	. . . . 5 |- ( W e. X -> V e. _V )
19	18	pwexd 4232	. . . 4 |- ( W e. X -> ~P V e. _V )
20	19	mptexd 5823	. . 3 |- ( W e. X -> ( s e. ~P V |-> |^| { t e. S | s C_ t } ) e. _V )
21	2, 12, 13, 20	fvmptd3 5685	. 2 |- ( W e. X -> ( LSpan ` W ) = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )
22	1, 21	eqtrid 2251	1 |- ( W e. X -> N = ( s e. ~P V |-> |^| { t e. S | s C_ t } ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   {crab 2489   _Vcvv 2773   C_ wss 3170   ~Pcpw 3620   |^|cint 3890   |-> cmpt 4112   Fn wfn 5274   ` cfv 5279   Basecbs 12902   LSubSpclss 14184   LSpanclspn 14218

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-cnex 8031  ax-resscn 8032  ax-1re 8034  ax-addrcl 8037

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-inn 9052  df-ndx 12905  df-slot 12906  df-base 12908  df-lsp 14219

This theorem is referenced by:  lspf  14221  lspval  14222  lspex  14227  lsppropd  14264