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Theorem hlrelat5N 28757
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 28756 . . 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 hllat 28720 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
8 id 21 . . . . . . . 8  |-  ( X  e.  B  ->  X  e.  B )
91, 4atbase 28646 . . . . . . . 8  |-  ( p  e.  A  ->  p  e.  B )
10 ovex 5817 . . . . . . . . . . . 12  |-  ( X 
.\/  p )  e. 
_V
1110a1i 12 . . . . . . . . . . 11  |-  ( p  e.  B  ->  ( X  .\/  p )  e. 
_V )
122, 3pltval 14056 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( X  .\/  p )  e.  _V )  -> 
( X  .<  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1311, 12syl3an3 1222 . . . . . . . . . 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 14128 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  X  .<_  ( X  .\/  p ) )
1615biantrurd 496 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  =/=  ( X  .\/  p )  <->  ( X  .<_  ( X  .\/  p
)  /\  X  =/=  ( X  .\/  p ) ) ) )
1713, 16bitr4d 249 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  X  =/=  ( X  .\/  p ) ) )
181, 2, 14latleeqj1 14131 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  p  e.  B  /\  X  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
19183com23 1162 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( p  .\/  X )  =  X ) )
201, 14latjcom 14127 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .\/  p
)  =  ( p 
.\/  X ) )
2120eqeq1d 2266 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( ( X  .\/  p )  =  X  <-> 
( p  .\/  X
)  =  X ) )
2219, 21bitr4d 249 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( p  .<_  X  <->  ( X  .\/  p )  =  X ) )
2322notbid 287 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <->  -.  ( X  .\/  p
)  =  X ) )
24 necom 2502 . . . . . . . . . . 11  |-  ( X  =/=  ( X  .\/  p )  <->  ( X  .\/  p )  =/=  X
)
25 df-ne 2423 . . . . . . . . . . 11  |-  ( ( X  .\/  p )  =/=  X  <->  -.  ( X  .\/  p )  =  X )
2624, 25bitri 242 . . . . . . . . . 10  |-  ( X  =/=  ( X  .\/  p )  <->  -.  ( X  .\/  p )  =  X )
2723, 26syl6bbr 256 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( -.  p  .<_  X  <-> 
X  =/=  ( X 
.\/  p ) ) )
2817, 27bitr4d 249 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  p  e.  B )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
297, 8, 9, 28syl3an 1229 . . . . . . 7  |-  ( ( K  e.  HL  /\  X  e.  B  /\  p  e.  A )  ->  ( X  .<  ( X  .\/  p )  <->  -.  p  .<_  X ) )
30293expa 1156 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( X  .<  ( X  .\/  p
)  <->  -.  p  .<_  X ) )
3130anbi1d 688 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B )  /\  p  e.  A
)  ->  ( ( X  .<  ( X  .\/  p )  /\  p  .<_  Y )  <->  ( -.  p  .<_  X  /\  p  .<_  Y ) ) )
3231rexbidva 2535 . . . 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 980 . . 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 453 . 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 225 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   _Vcvv 2763   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   ltcplt 14037   joincjn 14040   Latclat 14113   Atomscatm 28620   HLchlt 28707
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708
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