Theorem lsssetm 14665

Description: The set of all (not necessarily closed) linear subspaces of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 15-Jul-2014.)

Hypotheses

Ref	Expression
lssset.f	|- F = (Scalar ` W )
lssset.b	|- B = ( Base ` F )
lssset.v	|- V = ( Base ` W )
lssset.p	|- .+ = ( +g ` W )
lssset.t	|- .x. = ( .s ` W )
lssset.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lsssetm	|- ( W e. X -> S = { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } )

Distinct variable groups:   .+ , s   x, s, B   V, s   a, b, s, x, W   .x. , s   j, a, b, s, x

Allowed substitution hints:   B( j, a, b)   .+ ( x, j, a, b)   S( x, j, s, a, b)   .x. ( x, j, a, b)   F( x, j, s, a, b)   V( x, j, a, b)   W( j)   X( x, j, s, a, b)

Proof of Theorem lsssetm

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lssset.s	. 2 |- S = ( LSubSp ` W )
2		df-lssm 14662	. . 3 |- LSubSp = ( w e. _V |-> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } )
3		fveq2 5690	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lssset.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2289	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3690	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5690	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
8		lssset.f	. . . . . . . . 9 |- F = (Scalar ` W )
9	7, 8	eqtr4di 2289	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = F )
10	9	fveq2d 5694	. . . . . . 7 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
11		lssset.b	. . . . . . 7 |- B = ( Base ` F )
12	10, 11	eqtr4di 2289	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = B )
13		fveq2 5690	. . . . . . . . . . . 12 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		lssset.t	. . . . . . . . . . . 12 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2289	. . . . . . . . . . 11 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 6092	. . . . . . . . . 10 |- ( w = W -> ( x ( .s ` w ) a ) = ( x .x. a ) )
17	16	oveq1d 6090	. . . . . . . . 9 |- ( w = W -> ( ( x ( .s ` w ) a ) ( +g ` w ) b ) = ( ( x .x. a ) ( +g ` w ) b ) )
18		fveq2 5690	. . . . . . . . . . 11 |- ( w = W -> ( +g ` w ) = ( +g ` W ) )
19		lssset.p	. . . . . . . . . . 11 |- .+ = ( +g ` W )
20	18, 19	eqtr4di 2289	. . . . . . . . . 10 |- ( w = W -> ( +g ` w ) = .+ )
21	20	oveqd 6092	. . . . . . . . 9 |- ( w = W -> ( ( x .x. a ) ( +g ` w ) b ) = ( ( x .x. a ) .+ b ) )
22	17, 21	eqtrd 2271	. . . . . . . 8 |- ( w = W -> ( ( x ( .s ` w ) a ) ( +g ` w ) b ) = ( ( x .x. a ) .+ b ) )
23	22	eleq1d 2307	. . . . . . 7 |- ( w = W -> ( ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s <-> ( ( x .x. a ) .+ b ) e. s ) )
24	23	2ralbidv 2574	. . . . . 6 |- ( w = W -> ( A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s <-> A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) )
25	12, 24	raleqbidv 2765	. . . . 5 |- ( w = W -> ( A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s <-> A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) )
26	25	anbi2d 468	. . . 4 |- ( w = W -> ( ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) <-> ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) ) )
27	6, 26	rabeqbidv 2816	. . 3 |- ( w = W -> { s e. ~P ( Base ` w ) | ( E. j j e. s /\ A. x e. ( Base ` (Scalar ` w ) ) A. a e. s A. b e. s ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s ) } = { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } )
28		elex 2833	. . 3 |- ( W e. X -> W e. _V )
29		basfn 13389	. . . . . . 7 |- Base Fn _V
30		funfvex 5707	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
31	30	funfni 5478	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
32	29, 28, 31	sylancr 418	. . . . . 6 |- ( W e. X -> ( Base ` W ) e. _V )
33	4, 32	eqeltrid 2325	. . . . 5 |- ( W e. X -> V e. _V )
34	33	pwexd 4313	. . . 4 |- ( W e. X -> ~P V e. _V )
35		rabexg 4274	. . . 4 |- ( ~P V e. _V -> { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } e. _V )
36	34, 35	syl 14	. . 3 |- ( W e. X -> { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } e. _V )
37	2, 27, 28, 36	fvmptd3 5793	. 2 |- ( W e. X -> ( LSubSp ` W ) = { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } )
38	1, 37	eqtrid 2283	1 |- ( W e. X -> S = { s e. ~P V | ( E. j j e. s /\ A. x e. B A. a e. s A. b e. s ( ( x .x. a ) .+ b ) e. s ) } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   {crab 2532   _Vcvv 2821   ~Pcpw 3685   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  Scalarcsca 13411   .scvsca 13412   LSubSpclss 14661

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 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-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem 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-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-lssm 14662

This theorem is referenced by:  islssm  14666  islssmg  14667