Theorem lssclg 13860

Description: Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)

Hypotheses

Ref	Expression
lsscl.f	|- F = (Scalar ` W )
lsscl.b	|- B = ( Base ` F )
lsscl.p	|- .+ = ( +g ` W )
lsscl.t	|- .x. = ( .s ` W )
lsscl.s	|- S = ( LSubSp ` W )

Assertion

Ref	Expression
lssclg	|- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )

Proof of Theorem lssclg

Dummy variables x a b j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1000	. . . 4 |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> U e. S )
2		lsscl.f	. . . . . 6 |- F = (Scalar ` W )
3		lsscl.b	. . . . . 6 |- B = ( Base ` F )
4		eqid 2193	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
5		lsscl.p	. . . . . 6 |- .+ = ( +g ` W )
6		lsscl.t	. . . . . 6 |- .x. = ( .s ` W )
7		lsscl.s	. . . . . 6 |- S = ( LSubSp ` W )
8	2, 3, 4, 5, 6, 7	islssmg 13854	. . . . 5 |- ( W e. C -> ( U e. S <-> ( U C_ ( Base ` W ) /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) ) )
9	8	3ad2ant1 1020	. . . 4 |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( U e. S <-> ( U C_ ( Base ` W ) /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) ) )
10	1, 9	mpbid 147	. . 3 |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( U C_ ( Base ` W ) /\ E. j j e. U /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
11	10	simp3d 1013	. 2 |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U )
12		oveq1 5925	. . . . . 6 |- ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
13	12	oveq1d 5933	. . . . 5 |- ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
14	13	eleq1d 2262	. . . 4 |- ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
15		oveq2 5926	. . . . . 6 |- ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
16	15	oveq1d 5933	. . . . 5 |- ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
17	16	eleq1d 2262	. . . 4 |- ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
18		oveq2 5926	. . . . 5 |- ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
19	18	eleq1d 2262	. . . 4 |- ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
20	14, 17, 19	rspc3v 2880	. . 3 |- ( ( Z e. B /\ X e. U /\ Y e. U ) -> ( A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U -> ( ( Z .x. X ) .+ Y ) e. U ) )
21	20	3ad2ant3 1022	. 2 |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U -> ( ( Z .x. X ) .+ Y ) e. U ) )
22	11, 21	mpd 13	1 |- ( ( W e. C /\ U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   A.wral 2472   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695  Scalarcsca 12698   .scvsca 12699   LSubSpclss 13848

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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

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 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-lssm 13849

This theorem is referenced by:  lssvacl  13861  lssvsubcl  13862  lssvscl  13871  islss3  13875  lssintclm  13880