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Theorem snatpsubN 29939
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 3759 . . . . . 6  |-  ( P  e.  A  ->  { P }  C_  A )
21adantl 452 . . . . 5  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  C_  A )
3 atllat 29490 . . . . . . . . . . . . . . 15  |-  ( K  e.  AtLat  ->  K  e.  Lat )
4 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( Base `  K )  =  (
Base `  K )
5 snpsub.a . . . . . . . . . . . . . . . 16  |-  A  =  ( Atoms `  K )
64, 5atbase 29479 . . . . . . . . . . . . . . 15  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
7 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( join `  K )  =  (
join `  K )
84, 7latjidm 14180 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K ) )  -> 
( P ( join `  K ) P )  =  P )
93, 6, 8syl2an 463 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  ( P ( join `  K
) P )  =  P )
109adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( P
( join `  K ) P )  =  P )
1110breq2d 4035 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  <->  r ( le `  K ) P ) )
12 eqid 2283 . . . . . . . . . . . . . . . 16  |-  ( le
`  K )  =  ( le `  K
)
1312, 5atcmp 29501 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  AtLat  /\  r  e.  A  /\  P  e.  A )  ->  (
r ( le `  K ) P  <->  r  =  P ) )
14133com23 1157 . . . . . . . . . . . . . 14  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  r  e.  A )  ->  (
r ( le `  K ) P  <->  r  =  P ) )
15143expa 1151 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) P  <->  r  =  P ) )
1615biimpd 198 . . . . . . . . . . . 12  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) P  ->  r  =  P ) )
1711, 16sylbid 206 . . . . . . . . . . 11  |-  ( ( ( K  e.  AtLat  /\  P  e.  A )  /\  r  e.  A
)  ->  ( r
( le `  K
) ( P (
join `  K ) P )  ->  r  =  P ) )
1817adantld 453 . . . . . . . . . 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 3655 . . . . . . . . . . . . 13  |-  ( p  e.  { P }  <->  p  =  P )
20 elsn 3655 . . . . . . . . . . . . 13  |-  ( q  e.  { P }  <->  q  =  P )
2119, 20anbi12i 678 . . . . . . . . . . . 12  |-  ( ( p  e.  { P }  /\  q  e.  { P } )  <->  ( p  =  P  /\  q  =  P ) )
2221anbi1i 676 . . . . . . . . . . 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 5867 . . . . . . . . . . . . 13  |-  ( ( p  =  P  /\  q  =  P )  ->  ( p ( join `  K ) q )  =  ( P (
join `  K ) P ) )
2423breq2d 4035 . . . . . . . . . . . 12  |-  ( ( p  =  P  /\  q  =  P )  ->  ( r ( le
`  K ) ( p ( join `  K
) q )  <->  r ( le `  K ) ( P ( join `  K
) P ) ) )
2524pm5.32i 618 . . . . . . . . . . 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 240 . . . . . . . . . 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 3655 . . . . . . . . . 10  |-  ( r  e.  { P }  <->  r  =  P )
2818, 26, 273imtr4g 261 . . . . . . . . 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 590 . . . . . . . 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 72 . . . . . . 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 2632 . . . . . 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 2634 . . . . 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 518 . . . 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 423 . . 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 29934 . . 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 225 . 2  |-  ( K  e.  AtLat  ->  ( P  e.  A  ->  { P }  e.  S )
)
3837imp 418 1  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  { P }  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {csn 3640   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   AtLatcal 29454   PSubSpcpsubsp 29685
This theorem is referenced by:  pointpsubN  29940  pclfinN  30089  pclfinclN  30139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-join 14110  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488  df-psubsp 29692
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