Theorem islssmd 14372

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 3265	. . 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 1251	. . . . . . . 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 2614	. . . . . 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 5643	. . . . . . 7 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` W ) ) )
13	10, 12	eqtrd 2264	. . . . . 6 |- ( ph -> B = ( Base ` (Scalar ` W ) ) )
14	13	eleq2d 2301	. . . . 5 |- ( ph -> ( x e. B <-> x e. ( Base ` (Scalar ` W ) ) ) )
15		islssd.p	. . . . . . . . 9 |- ( ph -> .+ = ( +g ` W ) )
16	15	oveqd 6034	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x .x. a ) ( +g ` W ) b ) )
17		islssd.t	. . . . . . . . . 10 |- ( ph -> .x. = ( .s ` W ) )
18	17	oveqd 6034	. . . . . . . . 9 |- ( ph -> ( x .x. a ) = ( x ( .s ` W ) a ) )
19	18	oveq1d 6032	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) ( +g ` W ) b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
20	16, 19	eqtrd 2264	. . . . . . 7 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
21	20	eleq1d 2300	. . . . . 6 |- ( ph -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) )
22	21	2ralbidv 2556	. . . . 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 2604	. . 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 2231	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
27		eqid 2231	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
28		eqid 2231	. . . . 5 |- ( Base ` W ) = ( Base ` W )
29		eqid 2231	. . . . 5 |- ( +g ` W ) = ( +g ` W )
30		eqid 2231	. . . . 5 |- ( .s ` W ) = ( .s ` W )
31		eqid 2231	. . . . 5 |- ( LSubSp ` W ) = ( LSubSp ` W )
32	26, 27, 28, 29, 30, 31	islssmg 14371	. . . 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 1206	. 2 |- ( ph -> U e. ( LSubSp ` W ) )
35		islssd.s	. 2 |- ( ph -> S = ( LSubSp ` W ) )
36	34, 35	eleqtrrd 2311	1 |- ( ph -> U e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   C_ wss 3200   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159  Scalarcsca 13162   .scvsca 13163   LSubSpclss 14365

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-lssm 14366

This theorem is referenced by:  lss1  14375  lsssn0  14383  islss3  14392  lss1d  14396  lssintclm  14397