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Theorem ispsubcl2N 30583
Description: Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapsubcl.b  |-  B  =  ( Base `  K
)
pmapsubcl.m  |-  M  =  ( pmap `  K
)
pmapsubcl.c  |-  C  =  ( PSubCl `  K )
Assertion
Ref Expression
ispsubcl2N  |-  ( K  e.  HL  ->  ( X  e.  C  <->  E. y  e.  B  X  =  ( M `  y ) ) )
Distinct variable groups:    y, B    y, K    y, M    y, X
Allowed substitution hint:    C( y)

Proof of Theorem ispsubcl2N
StepHypRef Expression
1 eqid 2435 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 eqid 2435 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
3 pmapsubcl.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 30573 . 2  |-  ( K  e.  HL  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
5 hlop 29999 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
65adantr 452 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  OP )
7 hlclat 29995 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CLat )
87adantr 452 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  CLat )
91, 2polssatN 30544 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )
10 pmapsubcl.b . . . . . . . . . . 11  |-  B  =  ( Base `  K
)
1110, 1atssbase 29927 . . . . . . . . . 10  |-  ( Atoms `  K )  C_  B
129, 11syl6ss 3352 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  B )
13 eqid 2435 . . . . . . . . . 10  |-  ( lub `  K )  =  ( lub `  K )
1410, 13clatlubcl 14528 . . . . . . . . 9  |-  ( ( K  e.  CLat  /\  (
( _|_ P `  K ) `  X
)  C_  B )  ->  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )
158, 12, 14syl2anc 643 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )
16 eqid 2435 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
1710, 16opoccl 29831 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B )
186, 15, 17syl2anc 643 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) )  e.  B )
1918ex 424 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( oc `  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B ) )
2019adantrd 455 . . . . 5  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  ->  ( ( oc `  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B ) )
21 pmapsubcl.m . . . . . . . . . 10  |-  M  =  ( pmap `  K
)
2213, 16, 1, 21, 2polval2N 30542 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( ( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )  -> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
239, 22syldan 457 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
2423ex 424 . . . . . . 7  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  ( M `  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) ) ) ) )
25 eqeq1 2441 . . . . . . . 8  |-  ( ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X  ->  ( ( ( _|_ P `  K
) `  ( ( _|_ P `  K ) `
 X ) )  =  ( M `  ( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) ) )  <->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) )
2625biimpcd 216 . . . . . . 7  |-  ( ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) )  ->  ( ( ( _|_ P `  K
) `  ( ( _|_ P `  K ) `
 X ) )  =  X  ->  X  =  ( M `  ( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) ) ) ) )
2724, 26syl6 31 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X  ->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) ) )
2827imp3a 421 . . . . 5  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  ->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) )
2920, 28jcad 520 . . . 4  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  ->  ( (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) )  e.  B  /\  X  =  ( M `  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) ) ) ) ) )
30 fveq2 5719 . . . . . 6  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( M `  y )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
3130eqeq2d 2446 . . . . 5  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( X  =  ( M `  y )  <->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) )
3231rspcev 3044 . . . 4  |-  ( ( ( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B  /\  X  =  ( M `  ( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) ) ) )  ->  E. y  e.  B  X  =  ( M `  y ) )
3329, 32syl6 31 . . 3  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  ->  E. y  e.  B  X  =  ( M `  y ) ) )
3410, 1, 21pmapssat 30395 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( M `  y
)  C_  ( Atoms `  K ) )
3510, 21, 22polpmapN 30549 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) )
36 sseq1 3361 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  <->  ( M `  y )  C_  ( Atoms `  K ) ) )
37 fveq2 5719 . . . . . . . . 9  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  X
)  =  ( ( _|_ P `  K
) `  ( M `  y ) ) )
3837fveq2d 5723 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) ) )
39 id 20 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  X  =  ( M `  y ) )
4038, 39eqeq12d 2449 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  (
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X  <-> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) ) )
4136, 40anbi12d 692 . . . . . 6  |-  ( X  =  ( M `  y )  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  <->  ( ( M `
 y )  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) ) ) )
4241biimprcd 217 . . . . 5  |-  ( ( ( M `  y
)  C_  ( Atoms `  K )  /\  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) )  ->  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  /\  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  X ) ) )
4334, 35, 42syl2anc 643 . . . 4  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
4443rexlimdva 2822 . . 3  |-  ( K  e.  HL  ->  ( E. y  e.  B  X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  /\  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  X ) ) )
4533, 44impbid 184 . 2  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  <->  E. y  e.  B  X  =  ( M `  y ) ) )
464, 45bitrd 245 1  |-  ( K  e.  HL  ->  ( X  e.  C  <->  E. y  e.  B  X  =  ( M `  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   ` cfv 5445   Basecbs 13457   occoc 13525   lubclub 14387   CLatccla 14524   OPcops 29809   Atomscatm 29900   HLchlt 29987   pmapcpmap 30133   _|_
PcpolN 30538   PSubClcpscN 30570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-psubsp 30139  df-pmap 30140  df-polarityN 30539  df-psubclN 30571
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