Theorem lssclg 13456

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 998	. . . 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 2177	. . . . . 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	islssm 13450	. . . . 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 1018	. . . 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 1011	. 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 5884	. . . . . 6 |- ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
13	12	oveq1d 5892	. . . . 5 |- ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
14	13	eleq1d 2246	. . . 4 |- ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
15		oveq2 5885	. . . . . 6 |- ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
16	15	oveq1d 5892	. . . . 5 |- ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
17	16	eleq1d 2246	. . . 4 |- ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
18		oveq2 5885	. . . . 5 |- ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
19	18	eleq1d 2246	. . . 4 |- ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
20	14, 17, 19	rspc3v 2859	. . 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 1020	. 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 978   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   C_ wss 3131   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538  Scalarcsca 12541   .scvsca 12542   LSubSpclss 13447

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-lssm 13448

This theorem is referenced by:  lssvacl  13457  lssvsubcl  13458  lssvscl  13467  islss3  13471  lssintclm  13476