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Theorem snatpsubN 30484
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 3934 . . . . . 6  |-  ( P  e.  A  ->  { P }  C_  A )
21adantl 453 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  C_  A )
3 atllat 30035 . . . . . . . . . . . . . . 15  |-  ( K  e.  AtLat  ->  K  e.  Lat )
4 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
5 snpsub.a . . . . . . . . . . . . . . . 16  |-  A  =  ( Atoms `  K )
64, 5atbase 30024 . . . . . . . . . . . . . . 15  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
7 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( join `  K )  =  (
join `  K )
84, 7latjidm 14495 . . . . . . . . . . . . . . 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 4216 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  <->  r ( le `  K ) P ) )
12 eqid 2435 . . . . . . . . . . . . . . . 16  |-  ( le
`  K )  =  ( le `  K
)
1312, 5atcmp 30046 . . . . . . . . . . . . . . 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 3821 . . . . . . . . . . . . 13  |-  ( p  e.  { P }  <->  p  =  P )
20 elsn 3821 . . . . . . . . . . . . 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 6082 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  q  =  P )  ->  ( p ( join `  K ) q )  =  ( P (
join `  K ) P ) )
2423breq2d 4216 . . . . . . . . . . . 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 3821 . . . . . . . . . 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 2787 . . . . . 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 2789 . . . . 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 30479 . . 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 1652    e. wcel 1725   A.wral 2697    C_ wss 3312   {csn 3806   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   Latclat 14466   Atomscatm 29998   AtLatcal 29999   PSubSpcpsubsp 30230
This theorem is referenced by:  pointpsubN  30485  pclfinN  30634  pclfinclN  30684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-join 14425  df-lat 14467  df-covers 30001  df-ats 30002  df-atl 30033  df-psubsp 30237
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