Theorem islpoldN 36773

Description: Properties that determine a polarity. (Contributed by NM, 26-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 )
islpold.w	|- ( ph -> W e. X )
islpold.1	|- ( ph -> ._|_ : ~P V --> S )
islpold.2	|- ( ph -> ( ._|_ ` V ) = { .0. } )
islpold.3	|- ( ( ph /\ ( x C_ V /\ y C_ V /\ x C_ y ) ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) )
islpold.4	|- ( ( ph /\ x e. A ) -> ( ._|_ ` x ) e. H )
islpold.5	|- ( ( ph /\ x e. A ) -> ( ._|_ ` ( ._|_ ` x ) ) = x )

Assertion

Ref	Expression
islpoldN	|- ( ph -> ._|_ e. P )

Distinct variable groups:   x, A   x, y, W   x, ._|_ , y   ph, x, y

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

Proof of Theorem islpoldN

Step	Hyp	Ref	Expression
1		islpold.1	. 2 |- ( ph -> ._|_ : ~P V --> S )
2		islpold.2	. . 3 |- ( ph -> ( ._|_ ` V ) = { .0. } )
3		islpold.3	. . . . 5 |- ( ( ph /\ ( x C_ V /\ y C_ V /\ x C_ y ) ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) )
4	3	ex 450	. . . 4 |- ( ph -> ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) )
5	4	alrimivv 1856	. . 3 |- ( ph -> A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) )
6		islpold.4	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ._|_ ` x ) e. H )
7		islpold.5	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ._|_ ` ( ._|_ ` x ) ) = x )
8	6, 7	jca 554	. . . 4 |- ( ( ph /\ x e. A ) -> ( ( ._|_ ` x ) e. H /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) )
9	8	ralrimiva 2966	. . 3 |- ( ph -> A. x e. A ( ( ._|_ ` x ) e. H /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) )
10	2, 5, 9	3jca 1242	. 2 |- ( ph -> ( ( ._|_ ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) /\ A. x e. A ( ( ._|_ ` x ) e. H /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) )
11		islpold.w	. . 3 |- ( ph -> W e. X )
12		lpolset.v	. . . 4 |- V = ( Base ` W )
13		lpolset.s	. . . 4 |- S = ( LSubSp ` W )
14		lpolset.z	. . . 4 |- .0. = ( 0g ` W )
15		lpolset.a	. . . 4 |- A = (LSAtoms ` W )
16		lpolset.h	. . . 4 |- H = (LSHyp ` W )
17		lpolset.p	. . . 4 |- P = (LPol ` W )
18	12, 13, 14, 15, 16, 17	islpolN 36772	. . 3 |- ( W e. X -> ( ._|_ e. P <-> ( ._|_ : ~P V --> S /\ ( ( ._|_ ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) /\ A. x e. A ( ( ._|_ ` x ) e. H /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) ) ) )
19	11, 18	syl 17	. 2 |- ( ph -> ( ._|_ e. P <-> ( ._|_ : ~P V --> S /\ ( ( ._|_ ` V ) = { .0. } /\ A. x A. y ( ( x C_ V /\ y C_ V /\ x C_ y ) -> ( ._|_ ` y ) C_ ( ._|_ ` x ) ) /\ A. x e. A ( ( ._|_ ` x ) e. H /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) ) ) )
20	1, 10, 19	mpbir2and 957	1 |- ( ph -> ._|_ e. P )

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:  dochpolN  36779