Theorem islssmd 13915

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 3221	. . 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 2579	. . . . . 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 5562	. . . . . . 7 |- ( ph -> ( Base ` F ) = ( Base ` (Scalar ` W ) ) )
13	10, 12	eqtrd 2229	. . . . . 6 |- ( ph -> B = ( Base ` (Scalar ` W ) ) )
14	13	eleq2d 2266	. . . . 5 |- ( ph -> ( x e. B <-> x e. ( Base ` (Scalar ` W ) ) ) )
15		islssd.p	. . . . . . . . 9 |- ( ph -> .+ = ( +g ` W ) )
16	15	oveqd 5939	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x .x. a ) ( +g ` W ) b ) )
17		islssd.t	. . . . . . . . . 10 |- ( ph -> .x. = ( .s ` W ) )
18	17	oveqd 5939	. . . . . . . . 9 |- ( ph -> ( x .x. a ) = ( x ( .s ` W ) a ) )
19	18	oveq1d 5937	. . . . . . . 8 |- ( ph -> ( ( x .x. a ) ( +g ` W ) b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
20	16, 19	eqtrd 2229	. . . . . . 7 |- ( ph -> ( ( x .x. a ) .+ b ) = ( ( x ( .s ` W ) a ) ( +g ` W ) b ) )
21	20	eleq1d 2265	. . . . . 6 |- ( ph -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( x ( .s ` W ) a ) ( +g ` W ) b ) e. U ) )
22	21	2ralbidv 2521	. . . . 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 2569	. . 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 2196	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
27		eqid 2196	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
28		eqid 2196	. . . . 5 |- ( Base ` W ) = ( Base ` W )
29		eqid 2196	. . . . 5 |- ( +g ` W ) = ( +g ` W )
30		eqid 2196	. . . . 5 |- ( .s ` W ) = ( .s ` W )
31		eqid 2196	. . . . 5 |- ( LSubSp ` W ) = ( LSubSp ` W )
32	26, 27, 28, 29, 30, 31	islssmg 13914	. . . 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 2276	1 |- ( ph -> U e. S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   C_ wss 3157   ` 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:  lss1  13918  lsssn0  13926  islss3  13935  lss1d  13939  lssintclm  13940