Theorem lssclg 14673

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 1029	. . . 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 2238	. . . . . 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 14667	. . . . 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 1049	. . . 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 1042	. 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 6082	. . . . . 6 |- ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
13	12	oveq1d 6090	. . . . 5 |- ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
14	13	eleq1d 2307	. . . 4 |- ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
15		oveq2 6083	. . . . . 6 |- ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
16	15	oveq1d 6090	. . . . 5 |- ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
17	16	eleq1d 2307	. . . 4 |- ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
18		oveq2 6083	. . . . 5 |- ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
19	18	eleq1d 2307	. . . 4 |- ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
20	14, 17, 19	rspc3v 2946	. . 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 1051	. 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 1009   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   C_ wss 3220   ` cfv 5372  (class class class)co 6075   Basecbs 13330   +g cplusg 13408  Scalarcsca 13411   .scvsca 13412   LSubSpclss 14661

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-lssm 14662

This theorem is referenced by:  lssvacl  14674  lssvsubcl  14675  lssvscl  14684  islss3  14688  lssintclm  14693