Theorem pol0N 35195

Description: The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
polssat.a	|- A = ( Atoms ` K )
polssat.p	|- ._|_ = ( _|_P ` K )

Assertion

Ref	Expression
pol0N	|- ( K e. B -> ( ._|_ ` (/) ) = A )

Proof of Theorem pol0N

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 3972	. . 3 |- (/) C_ A
2		eqid 2622	. . . 4 |- ( oc ` K ) = ( oc ` K )
3		polssat.a	. . . 4 |- A = ( Atoms ` K )
4		eqid 2622	. . . 4 |- ( pmap ` K ) = ( pmap ` K )
5		polssat.p	. . . 4 |- ._|_ = ( _|_P ` K )
6	2, 3, 4, 5	polvalN 35191	. . 3 |- ( ( K e. B /\ (/) C_ A ) -> ( ._|_ ` (/) ) = ( A i^i |^|_ p e. (/) ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
7	1, 6	mpan2 707	. 2 |- ( K e. B -> ( ._|_ ` (/) ) = ( A i^i |^|_ p e. (/) ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) )
8		0iin 4578	. . . 4 |- |^|_ p e. (/) ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) = _V
9	8	ineq2i 3811	. . 3 |- ( A i^i |^|_ p e. (/) ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) = ( A i^i _V )
10		inv1 3970	. . 3 |- ( A i^i _V ) = A
11	9, 10	eqtri 2644	. 2 |- ( A i^i |^|_ p e. (/) ( ( pmap ` K ) ` ( ( oc ` K ) ` p ) ) ) = A
12	7, 11	syl6eq 2672	1 |- ( K e. B -> ( ._|_ ` (/) ) = A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|_ciin 4521   ` cfv 5888   occoc 15949   Atomscatm 34550   pmapcpmap 34783   _|_PcpolN 35188

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-iin 4523  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-polarityN 35189

This theorem is referenced by:  2pol0N  35197  1psubclN  35230  osumcllem9N  35250  pexmidN  35255  pexmidlem6N  35261  pexmidALTN  35264