Theorem lpolsetN 36771

Description: The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lpolset.v	|- V = ( Base ` W )
lpolset.s	|- S = ( LSubSp ` W )
lpolset.z	|- .0. = ( 0g ` W )
lpolset.a	|- A = (LSAtoms ` W )
lpolset.h	|- H = (LSHyp ` W )
lpolset.p	|- P = (LPol ` W )

Assertion

Ref	Expression
lpolsetN	|- ( W e. X -> P = { o e. ( S ^m ~P V ) | ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) } )

Distinct variable groups:   x, A   S, o   o, V   x, o, y, W

Allowed substitution hints:   A( y, o)   P( x, y, o)   S( x, y)   H( x, y, o)   V( x, y)   X( x, y, o)   .0. ( x, y, o)

Proof of Theorem lpolsetN

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( W e. X -> W e. _V )
2		lpolset.p	. . 3 |- P = (LPol ` W )
3		fveq2 6191	. . . . . . 7 |- ( w = W -> ( LSubSp ` w ) = ( LSubSp ` W ) )
4		lpolset.s	. . . . . . 7 |- S = ( LSubSp ` W )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( LSubSp ` w ) = S )
6		fveq2 6191	. . . . . . . 8 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
7		lpolset.v	. . . . . . . 8 |- V = ( Base ` W )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( Base ` w ) = V )
9	8	pweqd 4163	. . . . . 6 |- ( w = W -> ~P ( Base ` w ) = ~P V )
10	5, 9	oveq12d 6668	. . . . 5 |- ( w = W -> ( ( LSubSp ` w ) ^m ~P ( Base ` w ) ) = ( S ^m ~P V ) )
11	8	fveq2d 6195	. . . . . . 7 |- ( w = W -> ( o ` ( Base ` w ) ) = ( o ` V ) )
12		fveq2 6191	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
13		lpolset.z	. . . . . . . . 9 |- .0. = ( 0g ` W )
14	12, 13	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> ( 0g ` w ) = .0. )
15	14	sneqd 4189	. . . . . . 7 |- ( w = W -> { ( 0g ` w ) } = { .0. } )
16	11, 15	eqeq12d 2637	. . . . . 6 |- ( w = W -> ( ( o ` ( Base ` w ) ) = { ( 0g ` w ) } <-> ( o ` V ) = { .0. } ) )
17	8	sseq2d 3633	. . . . . . . . 9 |- ( w = W -> ( x C_ ( Base ` w ) <-> x C_ V ) )
18	8	sseq2d 3633	. . . . . . . . 9 |- ( w = W -> ( y C_ ( Base ` w ) <-> y C_ V ) )
19	17, 18	3anbi12d 1400	. . . . . . . 8 |- ( w = W -> ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) <-> ( x C_ V /\ y C_ V /\ x C_ y ) ) )
20	19	imbi1d 331	. . . . . . 7 |- ( w = W -> ( ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) <-> ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) ) )
21	20	2albidv 1851	. . . . . 6 |- ( w = W -> ( A. x A. y ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) <-> A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) ) )
22		fveq2 6191	. . . . . . . 8 |- ( w = W -> (LSAtoms ` w ) = (LSAtoms ` W ) )
23		lpolset.a	. . . . . . . 8 |- A = (LSAtoms ` W )
24	22, 23	syl6eqr 2674	. . . . . . 7 |- ( w = W -> (LSAtoms ` w ) = A )
25		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> (LSHyp ` w ) = (LSHyp ` W ) )
26		lpolset.h	. . . . . . . . . 10 |- H = (LSHyp ` W )
27	25, 26	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> (LSHyp ` w ) = H )
28	27	eleq2d 2687	. . . . . . . 8 |- ( w = W -> ( ( o ` x ) e. (LSHyp ` w ) <-> ( o ` x ) e. H ) )
29	28	anbi1d 741	. . . . . . 7 |- ( w = W -> ( ( ( o ` x ) e. (LSHyp ` w ) /\ ( o ` ( o ` x ) ) = x ) <-> ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) )
30	24, 29	raleqbidv 3152	. . . . . 6 |- ( w = W -> ( A. x e. (LSAtoms ` w ) ( ( o ` x ) e. (LSHyp ` w ) /\ ( o ` ( o ` x ) ) = x ) <-> A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) )
31	16, 21, 30	3anbi123d 1399	. . . . 5 |- ( w = W -> ( ( ( o ` ( Base ` w ) ) = { ( 0g ` w ) } /\ A. x A. y ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. (LSAtoms ` w ) ( ( o ` x ) e. (LSHyp ` w ) /\ ( o ` ( o ` x ) ) = x ) ) <-> ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) ) )
32	10, 31	rabeqbidv 3195	. . . 4 |- ( w = W -> { o e. ( ( LSubSp ` w ) ^m ~P ( Base ` w ) ) | ( ( o ` ( Base ` w ) ) = { ( 0g ` w ) } /\ A. x A. y ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. (LSAtoms ` w ) ( ( o ` x ) e. (LSHyp ` w ) /\ ( o ` ( o ` x ) ) = x ) ) } = { o e. ( S ^m ~P V ) | ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) } )
33		df-lpolN 36770	. . . 4 |- LPol = ( w e. _V |-> { o e. ( ( LSubSp ` w ) ^m ~P ( Base ` w ) ) | ( ( o ` ( Base ` w ) ) = { ( 0g ` w ) } /\ A. x A. y ( ( x C_ ( Base ` w ) /\ y C_ ( Base ` w ) /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. (LSAtoms ` w ) ( ( o ` x ) e. (LSHyp ` w ) /\ ( o ` ( o ` x ) ) = x ) ) } )
34		ovex 6678	. . . . 5 |- ( S ^m ~P V ) e. _V
35	34	rabex 4813	. . . 4 |- { o e. ( S ^m ~P V ) | ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) } e. _V
36	32, 33, 35	fvmpt 6282	. . 3 |- ( W e. _V -> (LPol ` W ) = { o e. ( S ^m ~P V ) | ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) } )
37	2, 36	syl5eq 2668	. 2 |- ( W e. _V -> P = { o e. ( S ^m ~P V ) | ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) } )
38	1, 37	syl 17	1 |- ( W e. X -> P = { o e. ( S ^m ~P V ) | ( ( o ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( o ` y ) C_ ( o ` x ) ) /\ A. x e. A ( ( o ` x ) e. H /\ ( o ` ( o ` x ) ) = x ) ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   0gc0g 16100   LSubSpclss 18932  LSAtomsclsa 34261  LSHypclsh 34262  LPolclpoN 36769

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-lpolN 36770

This theorem is referenced by:  islpolN  36772