Theorem lsscl 18943

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
lsscl	|- ( ( U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )

Proof of Theorem lsscl

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

Step	Hyp	Ref	Expression
1		lsscl.f	. . . 4 |- F = (Scalar ` W )
2		lsscl.b	. . . 4 |- B = ( Base ` F )
3		eqid 2622	. . . 4 |- ( Base ` W ) = ( Base ` W )
4		lsscl.p	. . . 4 |- .+ = ( +g ` W )
5		lsscl.t	. . . 4 |- .x. = ( .s ` W )
6		lsscl.s	. . . 4 |- S = ( LSubSp ` W )
7	1, 2, 3, 4, 5, 6	islss 18935	. . 3 |- ( U e. S <-> ( U C_ ( Base ` W ) /\ U =/= (/) /\ A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U ) )
8	7	simp3bi 1078	. 2 |- ( U e. S -> A. x e. B A. a e. U A. b e. U ( ( x .x. a ) .+ b ) e. U )
9		oveq1 6657	. . . . 5 |- ( x = Z -> ( x .x. a ) = ( Z .x. a ) )
10	9	oveq1d 6665	. . . 4 |- ( x = Z -> ( ( x .x. a ) .+ b ) = ( ( Z .x. a ) .+ b ) )
11	10	eleq1d 2686	. . 3 |- ( x = Z -> ( ( ( x .x. a ) .+ b ) e. U <-> ( ( Z .x. a ) .+ b ) e. U ) )
12		oveq2 6658	. . . . 5 |- ( a = X -> ( Z .x. a ) = ( Z .x. X ) )
13	12	oveq1d 6665	. . . 4 |- ( a = X -> ( ( Z .x. a ) .+ b ) = ( ( Z .x. X ) .+ b ) )
14	13	eleq1d 2686	. . 3 |- ( a = X -> ( ( ( Z .x. a ) .+ b ) e. U <-> ( ( Z .x. X ) .+ b ) e. U ) )
15		oveq2 6658	. . . 4 |- ( b = Y -> ( ( Z .x. X ) .+ b ) = ( ( Z .x. X ) .+ Y ) )
16	15	eleq1d 2686	. . 3 |- ( b = Y -> ( ( ( Z .x. X ) .+ b ) e. U <-> ( ( Z .x. X ) .+ Y ) e. U ) )
17	11, 14, 16	rspc3v 3325	. 2 |- ( ( 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 ) )
18	8, 17	mpan9 486	1 |- ( ( U e. S /\ ( Z e. B /\ X e. U /\ Y e. U ) ) -> ( ( Z .x. X ) .+ Y ) e. U )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   LSubSpclss 18932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-lss 18933

This theorem is referenced by:  lssvsubcl  18944  lssvacl  18954  lssvscl  18955  islss3  18959  lssintcl  18964  lspsolvlem  19142  lbsextlem2  19159  isphld  19999