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Theorem 2atlt 29921
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 29772 . . 3  |-  ( P  e.  A  ->  P  e.  B )
4 eqid 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 2atomslt.s . . . 4  |-  .<  =  ( lt `  K )
6 eqid 2404 . . . 4  |-  ( join `  K )  =  (
join `  K )
71, 4, 5, 6, 2hlrelat 29884 . . 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 1233 . 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 985 . . . . . . . 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 1050 . . . . . . . . 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 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 ) )  ->  P  e.  A )
12 simp2 958 . . . . . . . . 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 2404 . . . . . . . . . 10  |-  (  <o  `  K )  =  ( 
<o  `  K )
145, 6, 2, 13atltcvr 29917 . . . . . . . . 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 1186 . . . . . . . 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 202 . . . . . . 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 29899 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  =/=  q  <->  P (  <o  `  K )
( P ( join `  K ) q ) ) )
1810, 11, 12, 17syl3anc 1184 . . . . . . 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 224 . . . . . 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 2650 . . . . 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 29919 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  q  e.  A  /\  P  e.  A )  ->  ( q  .<  (
q ( join `  K
) P )  <->  q  =/=  P ) )
2210, 12, 11, 21syl3anc 1184 . . . . . . . 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 224 . . . . . . 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 29846 . . . . . . . . 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 29772 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
28273ad2ant2 979 . . . . . . . 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 14443 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  =  ( q (
join `  K ) P ) )
3025, 26, 28, 29syl3anc 1184 . . . . . . 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 4198 . . . . . 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 986 . . . . . 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 29848 . . . . . . . 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 14434 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  e.  B )
3625, 26, 28, 35syl3anc 1184 . . . . . . 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 1052 . . . . . . 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 14383 . . . . . . 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 1186 . . . . . 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 661 . . . . 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 519 . . . 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 1152 . . 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 2776 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   Posetcpo 14352   ltcplt 14353   joincjn 14356   Latclat 14429    <o ccvr 29745   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  cdlemb  30276  lhpexle1  30490
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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