Theorem lssclg 14126

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 1001	. . . 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 2205	. . . . . 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 14120	. . . . 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 1021	. . . 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 1014	. 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 5951	. . . . . 6 |- ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
13	12	oveq1d 5959	. . . . 5 |- ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
14	13	eleq1d 2274	. . . 4 |- ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
15		oveq2 5952	. . . . . 6 |- ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
16	15	oveq1d 5959	. . . . 5 |- ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
17	16	eleq1d 2274	. . . 4 |- ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
18		oveq2 5952	. . . . 5 |- ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
19	18	eleq1d 2274	. . . 4 |- ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
20	14, 17, 19	rspc3v 2893	. . 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 1023	. 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 981   = wceq 1373   E.wex 1515   e. wcel 2176   A.wral 2484   C_ wss 3166   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909  Scalarcsca 12912   .scvsca 12913   LSubSpclss 14114

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-lssm 14115

This theorem is referenced by:  lssvacl  14127  lssvsubcl  14128  lssvscl  14137  islss3  14141  lssintclm  14146