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Theorem hlrelat5N 29590
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 29589 . . 3  |-  ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
65imp 418 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  E. p  e.  A  ( -.  p  .<_  X  /\  p  .<_  Y ) )
7 hllat 29553 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
8 id 19 . . . . . . . 8  |-  ( X  e.  B  ->  X  e.  B )
91, 4atbase 29479 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
10 ovex 5883 . . . . . . . . . . . 12  |-  ( X 
.\/  p )  e. 
_V
1110a1i 10 . . . . . . . . . . 11  |-  ( p  e.  B  ->  ( X  .\/  p )  e. 
_V )
122, 3pltval 14094 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  p )  e.  _V )  -> 
( X  .<  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1311, 12syl3an3 1217 . . . . . . . . . 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 14166 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  X  .<_  ( X  .\/  p ) )
1615biantrurd 494 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  =/=  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1713, 16bitr4d 247 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  X  =/=  ( X  .\/  p ) ) )
181, 2, 14latleeqj1 14169 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  X  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
19183com23 1157 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
201, 14latjcom 14165 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .\/  p
)  =  ( p 
.\/  X ) )
2120eqeq1d 2291 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( ( X  .\/  p )  =  X  <-> 
( p  .\/  X
)  =  X ) )
2219, 21bitr4d 247 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( X  .\/  p )  =  X ) )
2322notbid 285 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <->  -.  ( X  .\/  p
)  =  X ) )
24 necom 2527 . . . . . . . . . . 11  |-  ( X  =/=  ( X  .\/  p )  <->  ( X  .\/  p )  =/=  X
)
25 df-ne 2448 . . . . . . . . . . 11  |-  ( ( X  .\/  p )  =/=  X  <->  -.  ( X  .\/  p )  =  X )
2624, 25bitri 240 . . . . . . . . . 10  |-  ( X  =/=  ( X  .\/  p )  <->  -.  ( X  .\/  p )  =  X )
2723, 26syl6bbr 254 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <-> 
X  =/=  ( X 
.\/  p ) ) )
2817, 27bitr4d 247 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
297, 8, 9, 28syl3an 1224 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
30293expa 1151 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( X  .<  ( X  .\/  p
)  <->  -.  p  .<_  X ) )
3130anbi1d 685 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( ( X  .<  ( X  .\/  p )  /\  p  .<_  Y )  <->  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
3231rexbidva 2560 . . . 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 975 . . 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 451 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   _Vcvv 2788   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   ltcplt 14075   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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