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Theorem hlrelat 32892
Description: A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 28009 analog.) (Contributed by NM, 4-Feb-2012.)
Hypotheses
Ref Expression
hlrelat5.b  |-  B  =  ( Base `  K
)
hlrelat5.l  |-  .<_  =  ( le `  K )
hlrelat5.s  |-  .<  =  ( lt `  K )
hlrelat5.j  |-  .\/  =  ( join `  K )
hlrelat5.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
hlrelat  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( X  .<  ( X 
.\/  p )  /\  ( X  .\/  p ) 
.<_  Y ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    X, p    Y, p    .< , p
Allowed substitution hint:    .\/ ( p)

Proof of Theorem hlrelat
StepHypRef Expression
1 hlrelat5.b . . . 4  |-  B  =  ( Base `  K
)
2 hlrelat5.l . . . 4  |-  .<_  =  ( le `  K )
3 hlrelat5.s . . . 4  |-  .<  =  ( lt `  K )
4 hlrelat5.a . . . 4  |-  A  =  ( Atoms `  K )
51, 2, 3, 4hlrelat1 32890 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
65imp 431 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) )
7 simpll1 1045 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  K  e.  HL )
8 hllat 32854 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
97, 8syl 17 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  K  e.  Lat )
10 simpll2 1046 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  X  e.  B )
111, 4atbase 32780 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
1211adantl 468 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  p  e.  B )
13 hlrelat5.j . . . . . 6  |-  .\/  =  ( join `  K )
141, 2, 3, 13latnle 16324 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <-> 
X  .<  ( X  .\/  p ) ) )
159, 10, 12, 14syl3anc 1265 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  ( -.  p  .<_  X  <->  X  .<  ( X  .\/  p ) ) )
162, 3pltle 16200 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  .<_  Y ) )
1716imp 431 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  .<_  Y )
1817adantr 467 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  X  .<_  Y )
1918biantrurd 511 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
p  .<_  Y  <->  ( X  .<_  Y  /\  p  .<_  Y ) ) )
20 simpll3 1047 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  Y  e.  B )
211, 2, 13latjle12 16301 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  p  e.  B  /\  Y  e.  B
) )  ->  (
( X  .<_  Y  /\  p  .<_  Y )  <->  ( X  .\/  p )  .<_  Y ) )
229, 10, 12, 20, 21syl13anc 1267 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
( X  .<_  Y  /\  p  .<_  Y )  <->  ( X  .\/  p )  .<_  Y ) )
2319, 22bitrd 257 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
p  .<_  Y  <->  ( X  .\/  p )  .<_  Y ) )
2415, 23anbi12d 716 . . 3  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
( -.  p  .<_  X  /\  p  .<_  Y )  <-> 
( X  .<  ( X  .\/  p )  /\  ( X  .\/  p ) 
.<_  Y ) ) )
2524rexbidva 2937 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  ( E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y )  <->  E. p  e.  A  ( X  .<  ( X  .\/  p
)  /\  ( X  .\/  p )  .<_  Y ) ) )
266, 25mpbid 214 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( X  .<  ( X 
.\/  p )  /\  ( X  .\/  p ) 
.<_  Y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   E.wrex 2777   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Basecbs 15114   lecple 15190   ltcplt 16179   joincjn 16182   Latclat 16284   Atomscatm 32754   HLchlt 32841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-lat 16285  df-clat 16347  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842
This theorem is referenced by:  hlrelat2  32893  atle  32926  2atlt  32929
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