Theorem lsssetm 14118

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 14115	. . 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 5576	. . . . . 6 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
4		lssset.v	. . . . . 6 |- V = ( Base ` W )
5	3, 4	eqtr4di 2256	. . . . 5 |- ( w = W -> ( Base ` w ) = V )
6	5	pweqd 3621	. . . 4 |- ( w = W -> ~P ( Base ` w ) = ~P V )
7		fveq2 5576	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
8		lssset.f	. . . . . . . . 9 |- F = (Scalar ` W )
9	7, 8	eqtr4di 2256	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = F )
10	9	fveq2d 5580	. . . . . . 7 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` F ) )
11		lssset.b	. . . . . . 7 |- B = ( Base ` F )
12	10, 11	eqtr4di 2256	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = B )
13		fveq2 5576	. . . . . . . . . . . 12 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
14		lssset.t	. . . . . . . . . . . 12 |- .x. = ( .s ` W )
15	13, 14	eqtr4di 2256	. . . . . . . . . . 11 |- ( w = W -> ( .s ` w ) = .x. )
16	15	oveqd 5961	. . . . . . . . . 10 |- ( w = W -> ( x ( .s ` w ) a ) = ( x .x. a ) )
17	16	oveq1d 5959	. . . . . . . . 9 |- ( w = W -> ( ( x ( .s ` w ) a ) ( +g ` w ) b ) = ( ( x .x. a ) ( +g ` w ) b ) )
18		fveq2 5576	. . . . . . . . . . 11 |- ( w = W -> ( +g ` w ) = ( +g ` W ) )
19		lssset.p	. . . . . . . . . . 11 |- .+ = ( +g ` W )
20	18, 19	eqtr4di 2256	. . . . . . . . . 10 |- ( w = W -> ( +g ` w ) = .+ )
21	20	oveqd 5961	. . . . . . . . 9 |- ( w = W -> ( ( x .x. a ) ( +g ` w ) b ) = ( ( x .x. a ) .+ b ) )
22	17, 21	eqtrd 2238	. . . . . . . 8 |- ( w = W -> ( ( x ( .s ` w ) a ) ( +g ` w ) b ) = ( ( x .x. a ) .+ b ) )
23	22	eleq1d 2274	. . . . . . 7 |- ( w = W -> ( ( ( x ( .s ` w ) a ) ( +g ` w ) b ) e. s <-> ( ( x .x. a ) .+ b ) e. s ) )
24	23	2ralbidv 2530	. . . . . 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 2718	. . . . 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 2767	. . 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 2783	. . 3 |- ( W e. X -> W e. _V )
29		basfn 12890	. . . . . . 7 |- Base Fn _V
30		funfvex 5593	. . . . . . . 8 |- ( ( Fun Base /\ W e. dom Base ) -> ( Base ` W ) e. _V )
31	30	funfni 5376	. . . . . . 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 2292	. . . . 5 |- ( W e. X -> V e. _V )
34	33	pwexd 4225	. . . 4 |- ( W e. X -> ~P V e. _V )
35		rabexg 4187	. . . 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 5673	. 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 2250	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 1373   E.wex 1515   e. wcel 2176   A.wral 2484   {crab 2488   _Vcvv 2772   ~Pcpw 3616   Fn wfn 5266   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  Scalarcsca 12912   .scvsca 12913   LSubSpclss 14114

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-lssm 14115

This theorem is referenced by:  islssm  14119  islssmg  14120