Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ispsubcl2N Unicode version

Theorem ispsubcl2N 30136
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 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 eqid 2283 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
3 pmapsubcl.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 30126 . 2  |-  ( K  e.  HL  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
5 hlop 29552 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
65adantr 451 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  OP )
7 hlclat 29548 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CLat )
87adantr 451 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  CLat )
91, 2polssatN 30097 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )
10 pmapsubcl.b . . . . . . . . . . 11  |-  B  =  ( Base `  K
)
1110, 1atssbase 29480 . . . . . . . . . 10  |-  ( Atoms `  K )  C_  B
129, 11syl6ss 3191 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  B )
13 eqid 2283 . . . . . . . . . 10  |-  ( lub `  K )  =  ( lub `  K )
1410, 13clatlubcl 14217 . . . . . . . . 9  |-  ( ( K  e.  CLat  /\  (
( _|_ P `  K ) `  X
)  C_  B )  ->  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )
158, 12, 14syl2anc 642 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )
16 eqid 2283 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
1710, 16opoccl 29384 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B )
186, 15, 17syl2anc 642 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) )  e.  B )
1918ex 423 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( oc `  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B ) )
2019adantrd 454 . . . . 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 30095 . . . . . . . . 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 456 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
2423ex 423 . . . . . . 7  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  ( M `  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) ) ) ) )
25 eqeq1 2289 . . . . . . . 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 215 . . . . . . 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 29 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X  ->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) ) )
2827imp3a 420 . . . . 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 519 . . . 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 5525 . . . . . 6  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( M `  y )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
3130eqeq2d 2294 . . . . 5  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( X  =  ( M `  y )  <->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) )
3231rspcev 2884 . . . 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 29 . . 3  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  ->  E. y  e.  B  X  =  ( M `  y ) ) )
3410, 1, 21pmapssat 29948 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( M `  y
)  C_  ( Atoms `  K ) )
3510, 21, 22polpmapN 30102 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) )
36 sseq1 3199 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  <->  ( M `  y )  C_  ( Atoms `  K ) ) )
37 fveq2 5525 . . . . . . . . 9  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  X
)  =  ( ( _|_ P `  K
) `  ( M `  y ) ) )
3837fveq2d 5529 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) ) )
39 id 19 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  X  =  ( M `  y ) )
4038, 39eqeq12d 2297 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  (
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X  <-> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) ) )
4136, 40anbi12d 691 . . . . . 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 216 . . . . 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 642 . . . 4  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
4443rexlimdva 2667 . . 3  |-  ( K  e.  HL  ->  ( E. y  e.  B  X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  /\  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  X ) ) )
4533, 44impbid 183 . 2  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  <->  E. y  e.  B  X  =  ( M `  y ) ) )
464, 45bitrd 244 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   ` cfv 5255   Basecbs 13148   occoc 13216   lubclub 14076   CLatccla 14213   OPcops 29362   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   _|_
PcpolN 30091   PSubClcpscN 30123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-polarityN 30092  df-psubclN 30124
  Copyright terms: Public domain W3C validator