Theorem lpolfN 36774

Description: Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lpolf.v	|- V = ( Base ` W )
lpolf.s	|- S = ( LSubSp ` W )
lpolf.p	|- P = (LPol ` W )
lpolf.w	|- ( ph -> W e. X )
lpolf.o	|- ( ph -> ._|_ e. P )

Assertion

Ref	Expression
lpolfN	|- ( ph -> ._|_ : ~P V --> S )

Proof of Theorem lpolfN

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpolf.o	. . 3 |- ( ph -> ._|_ e. P )
2		lpolf.w	. . . 4 |- ( ph -> W e. X )
3		lpolf.v	. . . . 5 |- V = ( Base ` W )
4		lpolf.s	. . . . 5 |- S = ( LSubSp ` W )
5		eqid 2622	. . . . 5 |- ( 0g ` W ) = ( 0g ` W )
6		eqid 2622	. . . . 5 |- (LSAtoms ` W ) = (LSAtoms ` W )
7		eqid 2622	. . . . 5 |- (LSHyp ` W ) = (LSHyp ` W )
8		lpolf.p	. . . . 5 |- P = (LPol ` W )
9	3, 4, 5, 6, 7, 8	islpolN 36772	. . . 4 |- ( W e. X -> ( ._|_ e. P <-> ( ._|_ : ~P V --> S /\ ( ( ._|_ ` V ) = { ( 0g ` W ) } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) /\ A. x e. (LSAtoms ` W ) ( ( ._|_ ` x ) e. (LSHyp ` W ) /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) ) ) )
10	2, 9	syl 17	. . 3 |- ( ph -> ( ._|_ e. P <-> ( ._|_ : ~P V --> S /\ ( ( ._|_ ` V ) = { ( 0g ` W ) } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) /\ A. x e. (LSAtoms ` W ) ( ( ._|_ ` x ) e. (LSHyp ` W ) /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) ) ) )
11	1, 10	mpbid 222	. 2 |- ( ph -> ( ._|_ : ~P V --> S /\ ( ( ._|_ ` V ) = { ( 0g ` W ) } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) /\ A. x e. (LSAtoms ` W ) ( ( ._|_ ` x ) e. (LSHyp ` W ) /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) ) )
12	11	simpld 475	1 |- ( ph -> ._|_ : ~P V --> S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   {csn 4177   -->wf 5884   ` cfv 5888   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-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  ax-un 6949

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-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-lpolN 36770

This theorem is referenced by: (None)