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Theorem hlrelat5N 29883
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 29882 . . 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 29846 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
8 id 20 . . . . . . . 8  |-  ( X  e.  B  ->  X  e.  B )
91, 4atbase 29772 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
10 ovex 6065 . . . . . . . . . . . 12  |-  ( X 
.\/  p )  e. 
_V
1110a1i 11 . . . . . . . . . . 11  |-  ( p  e.  B  ->  ( X  .\/  p )  e. 
_V )
122, 3pltval 14372 . . . . . . . . . . 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 14444 . . . . . . . . . . 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 14447 . . . . . . . . . . . . 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 14443 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .\/  p
)  =  ( p 
.\/  X ) )
2120eqeq1d 2412 . . . . . . . . . . . 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 2648 . . . . . . . . . . 11  |-  ( X  =/=  ( X  .\/  p )  <->  ( X  .\/  p )  =/=  X
)
25 df-ne 2569 . . . . . . . . . . 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 2683 . . . 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 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   _Vcvv 2916   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   ltcplt 14353   joincjn 14356   Latclat 14429   Atomscatm 29746   HLchlt 29833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834
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