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Theorem hlrelat5N 32885
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 32884 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
65imp 429 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) )
7 hllat 32848 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
8 id 22 . . . . . . . 8  |-  ( X  e.  B  ->  X  e.  B )
91, 4atbase 32774 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
10 ovex 6111 . . . . . . . . . . . 12  |-  ( X 
.\/  p )  e. 
_V
1110a1i 11 . . . . . . . . . . 11  |-  ( p  e.  B  ->  ( X  .\/  p )  e. 
_V )
122, 3pltval 15122 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  p )  e.  _V )  -> 
( X  .<  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1311, 12syl3an3 1253 . . . . . . . . . 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 15222 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  X  .<_  ( X  .\/  p ) )
1615biantrurd 508 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  =/=  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1713, 16bitr4d 256 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  X  =/=  ( X  .\/  p ) ) )
181, 2, 14latleeqj1 15225 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  X  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
19183com23 1193 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
201, 14latjcom 15221 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .\/  p
)  =  ( p 
.\/  X ) )
2120eqeq1d 2446 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( ( X  .\/  p )  =  X  <-> 
( p  .\/  X
)  =  X ) )
2219, 21bitr4d 256 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( X  .\/  p )  =  X ) )
2322notbid 294 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <->  -.  ( X  .\/  p
)  =  X ) )
24 necom 2688 . . . . . . . . . . 11  |-  ( X  =/=  ( X  .\/  p )  <->  ( X  .\/  p )  =/=  X
)
25 df-ne 2603 . . . . . . . . . . 11  |-  ( ( X  .\/  p )  =/=  X  <->  -.  ( X  .\/  p )  =  X )
2624, 25bitri 249 . . . . . . . . . 10  |-  ( X  =/=  ( X  .\/  p )  <->  -.  ( X  .\/  p )  =  X )
2723, 26syl6bbr 263 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <-> 
X  =/=  ( X 
.\/  p ) ) )
2817, 27bitr4d 256 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
297, 8, 9, 28syl3an 1260 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
30293expa 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( X  .<  ( X  .\/  p
)  <->  -.  p  .<_  X ) )
3130anbi1d 704 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( ( X  .<  ( X  .\/  p )  /\  p  .<_  Y )  <->  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
3231rexbidva 2727 . . . 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 1008 . . 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 465 . 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 232 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711   _Vcvv 2967   class class class wbr 4287   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   ltcplt 15103   joincjn 15106   Latclat 15207   Atomscatm 32748   HLchlt 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836
This theorem is referenced by: (None)
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