Theorem islssmd 14317

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 3262	. . 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 1249	. . . . . . . 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 2612	. . . . . 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 5630	. . . . . . 7 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` W ) ) )
13	10, 12	eqtrd 2262	. . . . . 6 |- ( ph -> B = ( Base ` (Scalar ` W ) ) )
14	13	eleq2d 2299	. . . . 5 |- ( ph -> ( x e. B <-> x e. ( Base ` (Scalar ` W ) ) ) )
15		islssd.p	. . . . . . . . 9 |- ( ph -> .+ = ( +g ` W ) )
16	15	oveqd 6017	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x .x. a ) ( +g ` W ) b ) )
17		islssd.t	. . . . . . . . . 10 |- ( ph -> .x. = ( .s ` W ) )
18	17	oveqd 6017	. . . . . . . . 9 |- ( ph -> ( x .x. a ) = ( x ( .s ` W ) a ) )
19	18	oveq1d 6015	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) ( +g ` W ) b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
20	16, 19	eqtrd 2262	. . . . . . 7 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
21	20	eleq1d 2298	. . . . . 6 |- ( ph -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) )
22	21	2ralbidv 2554	. . . . 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 2602	. . 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 2229	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
27		eqid 2229	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
28		eqid 2229	. . . . 5 |- ( Base ` W ) = ( Base ` W )
29		eqid 2229	. . . . 5 |- ( +g ` W ) = ( +g ` W )
30		eqid 2229	. . . . 5 |- ( .s ` W ) = ( .s ` W )
31		eqid 2229	. . . . 5 |- ( LSubSp ` W ) = ( LSubSp ` W )
32	26, 27, 28, 29, 30, 31	islssmg 14316	. . . 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 1204	. 2 |- ( ph -> U e. ( LSubSp ` W ) )
35		islssd.s	. 2 |- ( ph -> S = ( LSubSp ` W ) )
36	34, 35	eleqtrrd 2309	1 |- ( ph -> U e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   C_ wss 3197   ` cfv 5317  (class class class)co 6000   Basecbs 13027   +g cplusg 13105  Scalarcsca 13108   .scvsca 13109   LSubSpclss 14310

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-cnex 8086  ax-resscn 8087  ax-1re 8089  ax-addrcl 8092

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-ov 6003  df-inn 9107  df-ndx 13030  df-slot 13031  df-base 13033  df-lssm 14311

This theorem is referenced by:  lss1  14320  lsssn0  14328  islss3  14337  lss1d  14341  lssintclm  14342