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Theorem ispsubcl2N 28825
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 2253 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
2 eqid 2253 . . 3  |-  ( _|_
P `  K )  =  ( _|_ P `  K )
3 pmapsubcl.c . . 3  |-  C  =  ( PSubCl `  K )
41, 2, 3ispsubclN 28815 . 2  |-  ( K  e.  HL  ->  ( X  e.  C  <->  ( X  C_  ( Atoms `  K )  /\  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  X
) )  =  X ) ) )
5 hlop 28241 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  OP )
65adantr 453 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  OP )
7 hlclat 28237 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  CLat )
87adantr 453 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  K  e.  CLat )
91, 2polssatN 28786 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  ( Atoms `  K ) )
10 pmapsubcl.b . . . . . . . . . . 11  |-  B  =  ( Base `  K
)
1110, 1atssbase 28169 . . . . . . . . . 10  |-  ( Atoms `  K )  C_  B
129, 11syl6ss 3112 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  X  C_  ( Atoms `  K
) )  ->  (
( _|_ P `  K ) `  X
)  C_  B )
13 eqid 2253 . . . . . . . . . 10  |-  ( lub `  K )  =  ( lub `  K )
1410, 13clatlubcl 14061 . . . . . . . . 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 2253 . . . . . . . . 9  |-  ( oc
`  K )  =  ( oc `  K
)
1710, 16opoccl 28073 . . . . . . . 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 28784 . . . . . . . . 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 2259 . . . . . . . 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 5377 . . . . . 6  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( M `  y )  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) )
3130eqeq2d 2264 . . . . 5  |-  ( y  =  ( ( oc
`  K ) `  ( ( lub `  K
) `  ( ( _|_ P `  K ) `
 X ) ) )  ->  ( X  =  ( M `  y )  <->  X  =  ( M `  ( ( oc `  K ) `
 ( ( lub `  K ) `  (
( _|_ P `  K ) `  X
) ) ) ) ) )
3231rcla4ev 2821 . . . 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 28637 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( M `  y
)  C_  ( Atoms `  K ) )
3510, 21, 22polpmapN 28791 . . . . 5  |-  ( ( K  e.  HL  /\  y  e.  B )  ->  ( ( _|_ P `  K ) `  (
( _|_ P `  K ) `  ( M `  y )
) )  =  ( M `  y ) )
36 sseq1 3120 . . . . . . 7  |-  ( X  =  ( M `  y )  ->  ( X  C_  ( Atoms `  K
)  <->  ( M `  y )  C_  ( Atoms `  K ) ) )
37 fveq2 5377 . . . . . . . . 9  |-  ( X  =  ( M `  y )  ->  (
( _|_ P `  K ) `  X
)  =  ( ( _|_ P `  K
) `  ( M `  y ) ) )
3837fveq2d 5381 . . . . . . . 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 2267 . . . . . . 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 2629 . . 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 1619    e. wcel 1621   E.wrex 2510    C_ wss 3078   ` cfv 4592   Basecbs 13022   occoc 13090   lubclub 13920   CLatccla 14057   OPcops 28051   Atomscatm 28142   HLchlt 28229   pmapcpmap 28375   _|_
PcpolN 28780   PSubClcpscN 28812
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230  df-psubsp 28381  df-pmap 28382  df-polarityN 28781  df-psubclN 28813
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