Theorem lsssetm 13912

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 13909	. . 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 5558	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lssset.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2247	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3610	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5558	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
8		lssset.f	. . . . . . . . 9 |- F = (Scalar ` W )
9	7, 8	eqtr4di 2247	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = F )
10	9	fveq2d 5562	. . . . . . 7 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
11		lssset.b	. . . . . . 7 |- B = ( Base ` F )
12	10, 11	eqtr4di 2247	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = B )
13		fveq2 5558	. . . . . . . . . . . 12 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		lssset.t	. . . . . . . . . . . 12 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2247	. . . . . . . . . . 11 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 5939	. . . . . . . . . 10 |- ( w = W -> ( x ( .s ` w ) a ) = ( x .x. a ) )
17	16	oveq1d 5937	. . . . . . . . 9 |- ( w = W -> ( ( x ( .s ` w ) a ) ( +g ` w ) b ) = ( ( x .x. a ) ( +g ` w ) b ) )
18		fveq2 5558	. . . . . . . . . . 11 |- ( w = W -> ( +g ` w ) = ( +g ` W ) )
19		lssset.p	. . . . . . . . . . 11 |- .+ = ( +g ` W )
20	18, 19	eqtr4di 2247	. . . . . . . . . 10 |- ( w = W -> ( +g ` w ) = .+ )
21	20	oveqd 5939	. . . . . . . . 9 |- ( w = W -> ( ( x .x. a ) ( +g ` w ) b ) = ( ( x .x. a ) .+ b ) )
22	17, 21	eqtrd 2229	. . . . . . . 8 |- ( w = W -> ( ( x ( .s ` w ) a ) ( +g ` w ) b ) = ( ( x .x. a ) .+ b ) )
23	22	eleq1d 2265	. . . . . . 7 |- ( w = W -> ( ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s <-> ( ( x .x. a ) .+ b ) e. s ) )
24	23	2ralbidv 2521	. . . . . 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 2709	. . . . 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 464	. . . 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 2758	. . 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 2774	. . 3 |- ( W e. X -> W e. _V )
29		basfn 12736	. . . . . . 7 |- Base Fn _V
30		funfvex 5575	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
31	30	funfni 5358	. . . . . . 7 |- ( ( Base Fn _V /\ W e. _V ) -> ( Base ` W ) e. _V )
32	29, 28, 31	sylancr 414	. . . . . 6 |- ( W e. X -> ( Base ` W ) e. _V )
33	4, 32	eqeltrid 2283	. . . . 5 |- ( W e. X -> V e. _V )
34	33	pwexd 4214	. . . 4 |- ( W e. X -> ~P V e. _V )
35		rabexg 4176	. . . 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 5655	. 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 2241	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 1364   E.wex 1506   e. wcel 2167   A.wral 2475   {crab 2479   _Vcvv 2763   ~Pcpw 3605   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755  Scalarcsca 12758   .scvsca 12759   LSubSpclss 13908

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-lssm 13909

This theorem is referenced by:  islssm  13913  islssmg  13914