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Theorem 2atlt 30237
Description: Given an atom less than an element, there is another atom less than the element. (Contributed by NM, 6-May-2012.)
Hypotheses
Ref Expression
2atomslt.b  |-  B  =  ( Base `  K
)
2atomslt.s  |-  .<  =  ( lt `  K )
2atomslt.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atlt  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<  X ) )
Distinct variable groups:    A, q    B, q    K, q    P, q    .< , q    X, q

Proof of Theorem 2atlt
StepHypRef Expression
1 2atomslt.b . . . 4  |-  B  =  ( Base `  K
)
2 2atomslt.a . . . 4  |-  A  =  ( Atoms `  K )
31, 2atbase 30088 . . 3  |-  ( P  e.  A  ->  P  e.  B )
4 eqid 2437 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 2atomslt.s . . . 4  |-  .<  =  ( lt `  K )
6 eqid 2437 . . . 4  |-  ( join `  K )  =  (
join `  K )
71, 4, 5, 6, 2hlrelat 30200 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  B  /\  X  e.  B )  /\  P  .<  X )  ->  E. q  e.  A  ( P  .<  ( P ( join `  K
) q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )
83, 7syl3anl2 1234 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  E. q  e.  A  ( P  .<  ( P ( join `  K
) q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )
9 simp3l 986 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  P  .<  ( P (
join `  K )
q ) )
10 simp1l1 1051 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  K  e.  HL )
11 simp1l2 1052 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  P  e.  A )
12 simp2 959 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
q  e.  A )
13 eqid 2437 . . . . . . . . . 10  |-  (  <o  `  K )  =  ( 
<o  `  K )
145, 6, 2, 13atltcvr 30233 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  P  e.  A  /\  q  e.  A
) )  ->  ( P  .<  ( P (
join `  K )
q )  <->  P (  <o  `  K ) ( P ( join `  K
) q ) ) )
1510, 11, 11, 12, 14syl13anc 1187 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( P  .<  ( P ( join `  K
) q )  <->  P (  <o  `  K ) ( P ( join `  K
) q ) ) )
169, 15mpbid 203 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  P (  <o  `  K
) ( P (
join `  K )
q ) )
176, 13, 2atcvr1 30215 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  =/=  q  <->  P (  <o  `  K )
( P ( join `  K ) q ) ) )
1810, 11, 12, 17syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( P  =/=  q  <->  P (  <o  `  K )
( P ( join `  K ) q ) ) )
1916, 18mpbird 225 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  P  =/=  q )
2019necomd 2688 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
q  =/=  P )
215, 6, 2atlt 30235 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  q  e.  A  /\  P  e.  A )  ->  ( q  .<  (
q ( join `  K
) P )  <->  q  =/=  P ) )
2210, 12, 11, 21syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( q  .<  (
q ( join `  K
) P )  <->  q  =/=  P ) )
2320, 22mpbird 225 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
q  .<  ( q (
join `  K ) P ) )
24 hllat 30162 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
2510, 24syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  K  e.  Lat )
2611, 3syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  P  e.  B )
271, 2atbase 30088 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
28273ad2ant2 980 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
q  e.  B )
291, 6latjcom 14489 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  =  ( q (
join `  K ) P ) )
3025, 26, 28, 29syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( P ( join `  K ) q )  =  ( q (
join `  K ) P ) )
3123, 30breqtrrd 4239 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
q  .<  ( P (
join `  K )
q ) )
32 simp3r 987 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( P ( join `  K ) q ) ( le `  K
) X )
33 hlpos 30164 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Poset )
3410, 33syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  K  e.  Poset )
351, 6latjcl 14480 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  e.  B )
3625, 26, 28, 35syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( P ( join `  K ) q )  e.  B )
37 simp1l3 1053 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  ->  X  e.  B )
381, 4, 5pltletr 14429 . . . . . . 7  |-  ( ( K  e.  Poset  /\  (
q  e.  B  /\  ( P ( join `  K
) q )  e.  B  /\  X  e.  B ) )  -> 
( ( q  .< 
( P ( join `  K ) q )  /\  ( P (
join `  K )
q ) ( le
`  K ) X )  ->  q  .<  X ) )
3934, 28, 36, 37, 38syl13anc 1187 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( ( q  .< 
( P ( join `  K ) q )  /\  ( P (
join `  K )
q ) ( le
`  K ) X )  ->  q  .<  X ) )
4031, 32, 39mp2and 662 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
q  .<  X )
4120, 40jca 520 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  /\  q  e.  A  /\  ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X ) )  -> 
( q  =/=  P  /\  q  .<  X ) )
42413exp 1153 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  ( q  e.  A  ->  ( ( P  .<  ( P (
join `  K )
q )  /\  ( P ( join `  K
) q ) ( le `  K ) X )  ->  (
q  =/=  P  /\  q  .<  X ) ) ) )
4342reximdvai 2817 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  ( E. q  e.  A  ( P  .<  ( P ( join `  K ) q )  /\  ( P (
join `  K )
q ) ( le
`  K ) X )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<  X ) ) )
448, 43mpd 15 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  X  e.  B )  /\  P  .<  X )  ->  E. q  e.  A  ( q  =/=  P  /\  q  .<  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   Posetcpo 14398   ltcplt 14399   joincjn 14402   Latclat 14475    <o ccvr 30061   Atomscatm 30062   HLchlt 30149
This theorem is referenced by:  cdlemb  30592  lhpexle1  30806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150
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