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Theorem snatpsubN 33752
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 4128 . . . . . 6  |-  ( P  e.  A  ->  { P }  C_  A )
21adantl 466 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  C_  A )
3 atllat 33303 . . . . . . . . . . . . . . 15  |-  ( K  e.  AtLat  ->  K  e.  Lat )
4 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
5 snpsub.a . . . . . . . . . . . . . . . 16  |-  A  =  ( Atoms `  K )
64, 5atbase 33292 . . . . . . . . . . . . . . 15  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
7 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( join `  K )  =  (
join `  K )
84, 7latjidm 15366 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P ( join `  K ) P )  =  P )
93, 6, 8syl2an 477 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( P ( join `  K
) P )  =  P )
109adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( P
( join `  K ) P )  =  P )
1110breq2d 4415 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  <->  r ( le `  K ) P ) )
12 eqid 2454 . . . . . . . . . . . . . . . 16  |-  ( le
`  K )  =  ( le `  K
)
1312, 5atcmp 33314 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  AtLat  /\  r  e.  A  /\  P  e.  A )  ->  (
r ( le `  K ) P  <->  r  =  P ) )
14133com23 1194 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  r  e.  A )  ->  (
r ( le `  K ) P  <->  r  =  P ) )
15143expa 1188 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) P  <->  r  =  P ) )
1615biimpd 207 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) P  ->  r  =  P ) )
1711, 16sylbid 215 . . . . . . . . . . 11  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  ->  r  =  P ) )
1817adantld 467 . . . . . . . . . 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 4002 . . . . . . . . . . . . 13  |-  ( p  e.  { P }  <->  p  =  P )
20 elsn 4002 . . . . . . . . . . . . 13  |-  ( q  e.  { P }  <->  q  =  P )
2119, 20anbi12i 697 . . . . . . . . . . . 12  |-  ( ( p  e.  { P }  /\  q  e.  { P } )  <->  ( p  =  P  /\  q  =  P ) )
2221anbi1i 695 . . . . . . . . . . 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 6212 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  q  =  P )  ->  ( p ( join `  K ) q )  =  ( P (
join `  K ) P ) )
2423breq2d 4415 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  q  =  P )  ->  ( r ( le
`  K ) ( p ( join `  K
) q )  <->  r ( le `  K ) ( P ( join `  K
) P ) ) )
2524pm5.32i 637 . . . . . . . . . . 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 249 . . . . . . . . . 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 4002 . . . . . . . . . 10  |-  ( r  e.  { P }  <->  r  =  P )
2818, 26, 273imtr4g 270 . . . . . . . . 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 607 . . . . . . . 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 78 . . . . . . 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 2911 . . . . . 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 2913 . . . . 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 532 . . . 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 434 . . 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 33747 . . 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 234 . 2  |-  ( K  e.  AtLat  ->  ( P  e.  A  ->  { P }  e.  S )
)
3837imp 429 1  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2799    C_ wss 3439   {csn 3988   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   AtLatcal 33267   PSubSpcpsubsp 33498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-covers 33269  df-ats 33270  df-atl 33301  df-psubsp 33505
This theorem is referenced by:  pointpsubN  33753  pclfinN  33902  pclfinclN  33952
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