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Theorem ispsubcl2N 33687
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 2443 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 eqid 2443 . . 3  |-  ( _|_P `  K )  =  ( _|_P `  K )
3 pmapsubcl.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 33677 . 2  |-  ( K  e.  HL  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  X ) ) )
5 hlop 33103 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
65adantr 465 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  OP )
7 hlclat 33099 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CLat )
87adantr 465 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  CLat )
91, 2polssatN 33648 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_P `  K ) `  X
)  C_  ( Atoms `  K ) )
10 pmapsubcl.b . . . . . . . . . . 11  |-  B  =  ( Base `  K
)
1110, 1atssbase 33031 . . . . . . . . . 10  |-  ( Atoms `  K )  C_  B
129, 11syl6ss 3389 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_P `  K ) `  X
)  C_  B )
13 eqid 2443 . . . . . . . . . 10  |-  ( lub `  K )  =  ( lub `  K )
1410, 13clatlubcl 15303 . . . . . . . . 9  |-  ( ( K  e.  CLat  /\  (
( _|_P `  K ) `  X
)  C_  B )  ->  ( ( lub `  K
) `  ( ( _|_P `  K ) `
 X ) )  e.  B )
158, 12, 14syl2anc 661 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( lub `  K
) `  ( ( _|_P `  K ) `
 X ) )  e.  B )
16 eqid 2443 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
1710, 16opoccl 32935 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  ( ( _|_P `  K ) `
 X ) )  e.  B )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_P `  K ) `
 X ) ) )  e.  B )
186, 15, 17syl2anc 661 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_P `  K ) `  X
) ) )  e.  B )
1918ex 434 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( oc `  K ) `  ( ( lub `  K
) `  ( ( _|_P `  K ) `
 X ) ) )  e.  B ) )
2019adantrd 468 . . . . 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 33646 . . . . . . . . 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 470 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_P `  K ) `  X
) ) ) ) )
2423ex 434 . . . . . . 7  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 X ) )  =  ( M `  ( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_P `  K ) `
 X ) ) ) ) ) )
25 eqeq1 2449 . . . . . . . 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 224 . . . . . . 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 33 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( (
( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  X  ->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_P `  K ) `  X
) ) ) ) ) ) )
2827impd 431 . . . . 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 533 . . . 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 5712 . . . . . 6  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_P `  K ) `
 X ) ) )  ->  ( M `  y )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_P `  K ) `  X
) ) ) ) )
3130eqeq2d 2454 . . . . 5  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_P `  K ) `
 X ) ) )  ->  ( X  =  ( M `  y )  <->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_P `  K ) `  X
) ) ) ) ) )
3231rspcev 3094 . . . 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 33 . . 3  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  X )  ->  E. y  e.  B  X  =  ( M `  y ) ) )
3410, 1, 21pmapssat 33499 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( M `  y
)  C_  ( Atoms `  K ) )
3510, 21, 22polpmapN 33653 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  y )
) )  =  ( M `  y ) )
36 sseq1 3398 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  <->  ( M `  y )  C_  ( Atoms `  K ) ) )
37 fveq2 5712 . . . . . . . . 9  |-  ( X  =  ( M `  y )  ->  (
( _|_P `  K ) `  X
)  =  ( ( _|_P `  K
) `  ( M `  y ) ) )
3837fveq2d 5716 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  (
( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  y )
) ) )
39 id 22 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  X  =  ( M `  y ) )
4038, 39eqeq12d 2457 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  (
( ( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  X  <-> 
( ( _|_P `  K ) `  (
( _|_P `  K ) `  ( M `  y )
) )  =  ( M `  y ) ) )
4136, 40anbi12d 710 . . . . . 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 225 . . . . 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 661 . . . 4  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  X ) ) )
4443rexlimdva 2862 . . 3  |-  ( K  e.  HL  ->  ( E. y  e.  B  X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  /\  ( ( _|_P `  K ) `
 ( ( _|_P `  K ) `
 X ) )  =  X ) ) )
4533, 44impbid 191 . 2  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_P `  K ) `  (
( _|_P `  K ) `  X
) )  =  X )  <->  E. y  e.  B  X  =  ( M `  y ) ) )
464, 45bitrd 253 1  |-  ( K  e.  HL  ->  ( X  e.  C  <->  E. y  e.  B  X  =  ( M `  y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737    C_ wss 3349   ` cfv 5439   Basecbs 14195   occoc 14267   lubclub 15133   CLatccla 15298   OPcops 32913   Atomscatm 33004   HLchlt 33091   pmapcpmap 33237   _|_PcpolN 33642   PSubClcpscN 33674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-undef 6813  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-psubsp 33243  df-pmap 33244  df-polarityN 33643  df-psubclN 33675
This theorem is referenced by: (None)
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