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Theorem snatpsubN 29864
Description: The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
snpsub.a  |-  A  =  ( Atoms `  K )
snpsub.s  |-  S  =  ( PSubSp `  K )
Assertion
Ref Expression
snatpsubN  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  e.  S )

Proof of Theorem snatpsubN
Dummy variables  q  p  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snssi 3885 . . . . . 6  |-  ( P  e.  A  ->  { P }  C_  A )
21adantl 453 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  C_  A )
3 atllat 29415 . . . . . . . . . . . . . . 15  |-  ( K  e.  AtLat  ->  K  e.  Lat )
4 eqid 2387 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
5 snpsub.a . . . . . . . . . . . . . . . 16  |-  A  =  ( Atoms `  K )
64, 5atbase 29404 . . . . . . . . . . . . . . 15  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
7 eqid 2387 . . . . . . . . . . . . . . . 16  |-  ( join `  K )  =  (
join `  K )
84, 7latjidm 14430 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P ( join `  K ) P )  =  P )
93, 6, 8syl2an 464 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( P ( join `  K
) P )  =  P )
109adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( P
( join `  K ) P )  =  P )
1110breq2d 4165 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  <->  r ( le `  K ) P ) )
12 eqid 2387 . . . . . . . . . . . . . . . 16  |-  ( le
`  K )  =  ( le `  K
)
1312, 5atcmp 29426 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  AtLat  /\  r  e.  A  /\  P  e.  A )  ->  (
r ( le `  K ) P  <->  r  =  P ) )
14133com23 1159 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  r  e.  A )  ->  (
r ( le `  K ) P  <->  r  =  P ) )
15143expa 1153 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) P  <->  r  =  P ) )
1615biimpd 199 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) P  ->  r  =  P ) )
1711, 16sylbid 207 . . . . . . . . . . 11  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  ->  r  =  P ) )
1817adantld 454 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( (
( p  =  P  /\  q  =  P )  /\  r ( le `  K ) ( P ( join `  K ) P ) )  ->  r  =  P ) )
19 elsn 3772 . . . . . . . . . . . . 13  |-  ( p  e.  { P }  <->  p  =  P )
20 elsn 3772 . . . . . . . . . . . . 13  |-  ( q  e.  { P }  <->  q  =  P )
2119, 20anbi12i 679 . . . . . . . . . . . 12  |-  ( ( p  e.  { P }  /\  q  e.  { P } )  <->  ( p  =  P  /\  q  =  P ) )
2221anbi1i 677 . . . . . . . . . . 11  |-  ( ( ( p  e.  { P }  /\  q  e.  { P } )  /\  r ( le
`  K ) ( p ( join `  K
) q ) )  <-> 
( ( p  =  P  /\  q  =  P )  /\  r
( le `  K
) ( p (
join `  K )
q ) ) )
23 oveq12 6029 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  q  =  P )  ->  ( p ( join `  K ) q )  =  ( P (
join `  K ) P ) )
2423breq2d 4165 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  q  =  P )  ->  ( r ( le
`  K ) ( p ( join `  K
) q )  <->  r ( le `  K ) ( P ( join `  K
) P ) ) )
2524pm5.32i 619 . . . . . . . . . . 11  |-  ( ( ( p  =  P  /\  q  =  P )  /\  r ( le `  K ) ( p ( join `  K ) q ) )  <->  ( ( p  =  P  /\  q  =  P )  /\  r
( le `  K
) ( P (
join `  K ) P ) ) )
2622, 25bitri 241 . . . . . . . . . 10  |-  ( ( ( p  e.  { P }  /\  q  e.  { P } )  /\  r ( le
`  K ) ( p ( join `  K
) q ) )  <-> 
( ( p  =  P  /\  q  =  P )  /\  r
( le `  K
) ( P (
join `  K ) P ) ) )
27 elsn 3772 . . . . . . . . . 10  |-  ( r  e.  { P }  <->  r  =  P )
2818, 26, 273imtr4g 262 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( (
( p  e.  { P }  /\  q  e.  { P } )  /\  r ( le
`  K ) ( p ( join `  K
) q ) )  ->  r  e.  { P } ) )
2928exp4b 591 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  (
r  e.  A  -> 
( ( p  e. 
{ P }  /\  q  e.  { P } )  ->  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  { P } ) ) ) )
3029com23 74 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  (
( p  e.  { P }  /\  q  e.  { P } )  ->  ( r  e.  A  ->  ( r
( le `  K
) ( p (
join `  K )
q )  ->  r  e.  { P } ) ) ) )
3130ralrimdv 2738 . . . . . 6  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  (
( p  e.  { P }  /\  q  e.  { P } )  ->  A. r  e.  A  ( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  { P } ) ) )
3231ralrimivv 2740 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  A. p  e.  { P } A. q  e.  { P } A. r  e.  A  ( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  { P } ) )
332, 32jca 519 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( { P }  C_  A  /\  A. p  e.  { P } A. q  e. 
{ P } A. r  e.  A  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  { P } ) ) )
3433ex 424 . . 3  |-  ( K  e.  AtLat  ->  ( P  e.  A  ->  ( { P }  C_  A  /\  A. p  e.  { P } A. q  e. 
{ P } A. r  e.  A  (
r ( le `  K ) ( p ( join `  K
) q )  -> 
r  e.  { P } ) ) ) )
35 snpsub.s . . . 4  |-  S  =  ( PSubSp `  K )
3612, 7, 5, 35ispsubsp 29859 . . 3  |-  ( K  e.  AtLat  ->  ( { P }  e.  S  <->  ( { P }  C_  A  /\  A. p  e. 
{ P } A. q  e.  { P } A. r  e.  A  ( r ( le
`  K ) ( p ( join `  K
) q )  -> 
r  e.  { P } ) ) ) )
3734, 36sylibrd 226 . 2  |-  ( K  e.  AtLat  ->  ( P  e.  A  ->  { P }  e.  S )
)
3837imp 419 1  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    C_ wss 3263   {csn 3757   class class class wbr 4153   ` cfv 5394  (class class class)co 6020   Basecbs 13396   lecple 13463   joincjn 14328   Latclat 14401   Atomscatm 29378   AtLatcal 29379   PSubSpcpsubsp 29610
This theorem is referenced by:  pointpsubN  29865  pclfinN  30014  pclfinclN  30064
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-poset 14330  df-plt 14342  df-lub 14358  df-join 14360  df-lat 14402  df-covers 29381  df-ats 29382  df-atl 29413  df-psubsp 29617
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