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Theorem ispsubcl2N 29403
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 2284 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 eqid 2284 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
3 pmapsubcl.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 29393 . 2  |-  ( K  e.  HL  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
5 hlop 28819 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
65adantr 453 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  OP )
7 hlclat 28815 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CLat )
87adantr 453 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  CLat )
91, 2polssatN 29364 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )
10 pmapsubcl.b . . . . . . . . . . 11  |-  B  =  ( Base `  K
)
1110, 1atssbase 28747 . . . . . . . . . 10  |-  ( Atoms `  K )  C_  B
129, 11syl6ss 3192 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  B )
13 eqid 2284 . . . . . . . . . 10  |-  ( lub `  K )  =  ( lub `  K )
1410, 13clatlubcl 14211 . . . . . . . . 9  |-  ( ( K  e.  CLat  /\  (
( _|_ P `  K ) `  X
)  C_  B )  ->  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )
158, 12, 14syl2anc 645 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )
16 eqid 2284 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
1710, 16opoccl 28651 . . . . . . . 8  |-  ( ( K  e.  OP  /\  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) )  e.  B )  -> 
( ( oc `  K ) `  (
( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B )
186, 15, 17syl2anc 645 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) )  e.  B )
1918ex 425 . . . . . 6  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( oc `  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  e.  B ) )
2019adantrd 456 . . . . 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 29362 . . . . . . . . 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 458 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
2423ex 425 . . . . . . 7  |-  ( K  e.  HL  ->  ( X  C_  ( Atoms `  K
)  ->  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  ( M `  (
( oc `  K
) `  ( ( lub `  K ) `  ( ( _|_ P `  K ) `  X
) ) ) ) ) )
25 eqeq1 2290 . . . . . . . 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 217 . . . . . . 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 422 . . . . 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 521 . . . 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 5485 . . . . . 6  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( M `  y )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
3130eqeq2d 2295 . . . . 5  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( X  =  ( M `  y )  <->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) )
3231rspcev 2885 . . . 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 29215 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( M `  y
)  C_  ( Atoms `  K ) )
3510, 21, 22polpmapN 29369 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) )
36 sseq1 3200 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  <->  ( M `  y )  C_  ( Atoms `  K ) ) )
37 fveq2 5485 . . . . . . . . 9  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  X
)  =  ( ( _|_ P `  K
) `  ( M `  y ) ) )
3837fveq2d 5489 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) ) )
39 id 21 . . . . . . . 8  |-  ( X  =  ( M `  y )  ->  X  =  ( M `  y ) )
4038, 39eqeq12d 2298 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  (
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X  <-> 
( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) ) )
4136, 40anbi12d 694 . . . . . 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 218 . . . . 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 645 . . . 4  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
4443rexlimdva 2668 . . 3  |-  ( K  e.  HL  ->  ( E. y  e.  B  X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  /\  ( ( _|_ P `  K ) `
 ( ( _|_
P `  K ) `  X ) )  =  X ) ) )
4533, 44impbid 185 . 2  |-  ( K  e.  HL  ->  (
( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X )  <->  E. y  e.  B  X  =  ( M `  y ) ) )
464, 45bitrd 246 1  |-  ( K  e.  HL  ->  ( X  e.  C  <->  E. y  e.  B  X  =  ( M `  y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1628    e. wcel 1688   E.wrex 2545    C_ wss 3153   ` cfv 5221   Basecbs 13142   occoc 13210   lubclub 14070   CLatccla 14207   OPcops 28629   Atomscatm 28720   HLchlt 28807   pmapcpmap 28953   _|_
PcpolN 29358   PSubClcpscN 29390
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-p1 14140  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779  df-hlat 28808  df-psubsp 28959  df-pmap 28960  df-polarityN 29359  df-psubclN 29391
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