Theorem islssmd 13858

Description: Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)

Hypotheses

Ref	Expression
islssd.f	|- ( ph -> F = (Scalar ` W ) )
islssd.b	|- ( ph -> B = ( Base ` F ) )
islssd.v	|- ( ph -> V = ( Base ` W ) )
islssd.p	|- ( ph -> .+ = ( +g ` W ) )
islssd.t	|- ( ph -> .x. = ( .s ` W ) )
islssd.s	|- ( ph -> S = ( LSubSp ` W ) )
islssd.u	|- ( ph -> U C_ V )
islssmd.m	|- ( ph -> E. j j e. U )
islssd.c	|- ( ( ph /\ ( x e. B /\ a e. U /\ b e. U ) ) -> ( ( x .x. a ) .+ b ) e. U )
islssmd.w	|- ( ph -> W e. X )

Assertion

Ref	Expression
islssmd	|- ( ph -> U e. S )

Distinct variable groups:   a, b, x, ph   U, a, b, x   W, a, b, x   B, a, b   U, j, a, b, x

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

Proof of Theorem islssmd

Step	Hyp	Ref	Expression
1		islssd.u	. . . 4 |- ( ph -> U C_ V )
2		islssd.v	. . . 4 |- ( ph -> V = ( Base ` W ) )
3	1, 2	sseqtrd 3218	. . 3 |- ( ph -> U C_ ( Base ` W ) )
4		islssmd.m	. . 3 |- ( ph -> E. j j e. U )
5		islssd.c	. . . . . . . . 9 |- ( ( ph /\ ( x e. B /\ a e. U /\ b e. U ) ) -> ( ( x .x. a ) .+ b ) e. U )
6	5	3exp2 1227	. . . . . . . 8 |- ( ph -> ( x e. B -> ( a e. U -> ( b e. U -> ( ( x .x. a ) .+ b ) e. U ) ) ) )
7	6	imp43 355	. . . . . . 7 |- ( ( ( ph /\ x e. B ) /\ ( a e. U /\ b e. U ) ) -> ( ( x .x. a ) .+ b ) e. U )
8	7	ralrimivva 2576	. . . . . 6 |- ( ( ph /\ x e. B ) -> A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U )
9	8	ex 115	. . . . 5 |- ( ph -> ( x e. B -> A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
10		islssd.b	. . . . . . 7 |- ( ph -> B = ( Base ` F ) )
11		islssd.f	. . . . . . . 8 |- ( ph -> F = (Scalar ` W ) )
12	11	fveq2d 5559	. . . . . . 7 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` W ) ) )
13	10, 12	eqtrd 2226	. . . . . 6 |- ( ph -> B = ( Base ` (Scalar ` W ) ) )
14	13	eleq2d 2263	. . . . 5 |- ( ph -> ( x e. B <-> x e. ( Base ` (Scalar ` W ) ) ) )
15		islssd.p	. . . . . . . . 9 |- ( ph -> .+ = ( +g ` W ) )
16	15	oveqd 5936	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x .x. a ) ( +g ` W ) b ) )
17		islssd.t	. . . . . . . . . 10 |- ( ph -> .x. = ( .s ` W ) )
18	17	oveqd 5936	. . . . . . . . 9 |- ( ph -> ( x .x. a ) = ( x ( .s ` W ) a ) )
19	18	oveq1d 5934	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) ( +g ` W ) b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
20	16, 19	eqtrd 2226	. . . . . . 7 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
21	20	eleq1d 2262	. . . . . 6 |- ( ph -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) )
22	21	2ralbidv 2518	. . . . 5 |- ( ph -> ( A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U <-> A. a e. U A. b e. U ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) )
23	9, 14, 22	3imtr3d 202	. . . 4 |- ( ph -> ( x e. ( Base ` (Scalar ` W ) ) -> A. a e. U A. b e. U ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) )
24	23	ralrimiv 2566	. . 3 |- ( ph -> A. x e. ( Base ` (Scalar ` W ) ) A. a e. U A. b e. U ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U )
25		islssmd.w	. . . 4 |- ( ph -> W e. X )
26		eqid 2193	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
27		eqid 2193	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
28		eqid 2193	. . . . 5 |- ( Base ` W ) = ( Base ` W )
29		eqid 2193	. . . . 5 |- ( +g ` W ) = ( +g ` W )
30		eqid 2193	. . . . 5 |- ( .s ` W ) = ( .s ` W )
31		eqid 2193	. . . . 5 |- ( LSubSp ` W ) = ( LSubSp ` W )
32	26, 27, 28, 29, 30, 31	islssmg 13857	. . . 4 |- ( W e. X -> ( U e. ( LSubSp ` W ) <-> ( U C_ ( Base ` W ) /\ E. j j e. U /\ A. x e. ( Base ` (Scalar ` W ) ) A. a e. U A. b e. U ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) ) )
33	25, 32	syl 14	. . 3 |- ( ph -> ( U e. ( LSubSp ` W ) <-> ( U C_ ( Base ` W ) /\ E. j j e. U /\ A. x e. ( Base ` (Scalar ` W ) ) A. a e. U A. b e. U ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) ) )
34	3, 4, 24, 33	mpbir3and 1182	. 2 |- ( ph -> U e. ( LSubSp ` W ) )
35		islssd.s	. 2 |- ( ph -> S = ( LSubSp ` W ) )
36	34, 35	eleqtrrd 2273	1 |- ( ph -> U e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   C_ wss 3154   ` cfv 5255  (class class class)co 5919   Basecbs 12621   +g cplusg 12698  Scalarcsca 12701   .scvsca 12702   LSubSpclss 13851

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-lssm 13852

This theorem is referenced by:  lss1  13861  lsssn0  13869  islss3  13878  lss1d  13882  lssintclm  13883