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Theorem hlrelat5N 30037
Description: An atomistic lattice with 0 is relatively atomic, using the definition in Remark 2 of [Kalmbach] p. 149. (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)
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
hlrelat5N  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( X  .<  ( X 
.\/  p )  /\  p  .<_  Y ) )
Distinct variable groups:    A, p    B, p    K, p    .<_ , p    X, p    Y, p
Allowed substitution hints:    .< ( p)    .\/ ( p)

Proof of Theorem hlrelat5N
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 30036 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
65imp 419 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) )
7 hllat 30000 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
8 id 20 . . . . . . . 8  |-  ( X  e.  B  ->  X  e.  B )
91, 4atbase 29926 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
10 ovex 6097 . . . . . . . . . . . 12  |-  ( X 
.\/  p )  e. 
_V
1110a1i 11 . . . . . . . . . . 11  |-  ( p  e.  B  ->  ( X  .\/  p )  e. 
_V )
122, 3pltval 14405 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  p )  e.  _V )  -> 
( X  .<  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1311, 12syl3an3 1219 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
14 hlrelat5.j . . . . . . . . . . . 12  |-  .\/  =  ( join `  K )
151, 2, 14latlej1 14477 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  X  .<_  ( X  .\/  p ) )
1615biantrurd 495 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  =/=  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1713, 16bitr4d 248 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  X  =/=  ( X  .\/  p ) ) )
181, 2, 14latleeqj1 14480 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  X  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
19183com23 1159 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
201, 14latjcom 14476 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .\/  p
)  =  ( p 
.\/  X ) )
2120eqeq1d 2443 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( ( X  .\/  p )  =  X  <-> 
( p  .\/  X
)  =  X ) )
2219, 21bitr4d 248 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( X  .\/  p )  =  X ) )
2322notbid 286 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <->  -.  ( X  .\/  p
)  =  X ) )
24 necom 2679 . . . . . . . . . . 11  |-  ( X  =/=  ( X  .\/  p )  <->  ( X  .\/  p )  =/=  X
)
25 df-ne 2600 . . . . . . . . . . 11  |-  ( ( X  .\/  p )  =/=  X  <->  -.  ( X  .\/  p )  =  X )
2624, 25bitri 241 . . . . . . . . . 10  |-  ( X  =/=  ( X  .\/  p )  <->  -.  ( X  .\/  p )  =  X )
2723, 26syl6bbr 255 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <-> 
X  =/=  ( X 
.\/  p ) ) )
2817, 27bitr4d 248 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
297, 8, 9, 28syl3an 1226 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
30293expa 1153 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( X  .<  ( X  .\/  p
)  <->  -.  p  .<_  X ) )
3130anbi1d 686 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( ( X  .<  ( X  .\/  p )  /\  p  .<_  Y )  <->  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
3231rexbidva 2714 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  B )  ->  ( E. p  e.  A  ( X  .<  ( X  .\/  p )  /\  p  .<_  Y )  <->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
33323adant3 977 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( E. p  e.  A  ( X  .<  ( X  .\/  p )  /\  p  .<_  Y )  <->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
3433adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  ( E. p  e.  A  ( X  .<  ( X  .\/  p
)  /\  p  .<_  Y )  <->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
356, 34mpbird 224 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( X  .<  ( X 
.\/  p )  /\  p  .<_  Y ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   ltcplt 14386   joincjn 14389   Latclat 14462   Atomscatm 29900   HLchlt 29987
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 4692
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 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988
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