Theorem lssclg 13553

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 999	. . . 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 2187	. . . . . 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 13547	. . . . 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 1019	. . . 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 1012	. 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 5895	. . . . . 6 |- ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
13	12	oveq1d 5903	. . . . 5 |- ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
14	13	eleq1d 2256	. . . 4 |- ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
15		oveq2 5896	. . . . . 6 |- ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
16	15	oveq1d 5903	. . . . 5 |- ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
17	16	eleq1d 2256	. . . 4 |- ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
18		oveq2 5896	. . . . 5 |- ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
19	18	eleq1d 2256	. . . 4 |- ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
20	14, 17, 19	rspc3v 2869	. . 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 1021	. 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 979   = wceq 1363   E.wex 1502   e. wcel 2158   A.wral 2465   C_ wss 3141   ` cfv 5228  (class class class)co 5888   Basecbs 12476   +g cplusg 12551  Scalarcsca 12554   .scvsca 12555   LSubSpclss 13541

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-inn 8934  df-ndx 12479  df-slot 12480  df-base 12482  df-lssm 13542

This theorem is referenced by:  lssvacl  13554  lssvsubcl  13555  lssvscl  13564  islss3  13568  lssintclm  13573