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Theorem hlrelat 28280
Description: A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (chrelati 22774 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 28278 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
65imp 420 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) )
7 simpll1 999 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  K  e.  HL )
8 hllat 28242 . . . . . 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 1000 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  X  e.  B )
111, 4atbase 28168 . . . . . 6  |-  ( p  e.  A  ->  p  e.  B )
1211adantl 454 . . . . 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 14035 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <-> 
X  .<  ( X  .\/  p ) ) )
159, 10, 12, 14syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  ( -.  p  .<_  X  <->  X  .<  ( X  .\/  p ) ) )
162, 3pltle 13939 . . . . . . . 8  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  .<_  Y ) )
1716imp 420 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  .<_  Y )
1817adantr 453 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  X  .<_  Y )
1918biantrurd 496 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
p  .<_  Y  <->  ( X  .<_  Y  /\  p  .<_  Y ) ) )
20 simpll3 1001 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  Y  e.  B )
211, 2, 13latjle12 14012 . . . . . 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 1189 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
( X  .<_  Y  /\  p  .<_  Y )  <->  ( X  .\/  p )  .<_  Y ) )
2319, 22bitrd 246 . . . 4  |-  ( ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  /\  p  e.  A )  ->  (
p  .<_  Y  <->  ( X  .\/  p )  .<_  Y ) )
2415, 23anbi12d 694 . . 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 2524 . 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 203 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 set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   ltcplt 13919   joincjn 13922   Latclat 13995   Atomscatm 28142   HLchlt 28229
This theorem is referenced by:  hlrelat2  28281  atle  28314  2atlt  28317
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-lat 13996  df-clat 14058  df-oposet 28055  df-ol 28057  df-oml 28058  df-covers 28145  df-ats 28146  df-atl 28177  df-cvlat 28201  df-hlat 28230
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