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Theorem 2atlt 34253
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 34104 . . 3  |-  ( P  e.  A  ->  P  e.  B )
4 eqid 2467 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 2atomslt.s . . . 4  |-  .<  =  ( lt `  K )
6 eqid 2467 . . . 4  |-  ( join `  K )  =  (
join `  K )
71, 4, 5, 6, 2hlrelat 34216 . . 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 1277 . 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 1024 . . . . . . . 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 1089 . . . . . . . . 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 1090 . . . . . . . . 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 997 . . . . . . . . 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 2467 . . . . . . . . . 10  |-  (  <o  `  K )  =  ( 
<o  `  K )
145, 6, 2, 13atltcvr 34249 . . . . . . . . 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 1230 . . . . . . . 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 210 . . . . . . 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 34231 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  =/=  q  <->  P (  <o  `  K )
( P ( join `  K ) q ) ) )
1810, 11, 12, 17syl3anc 1228 . . . . . . 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 232 . . . . . 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 2738 . . . . 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 34251 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  q  e.  A  /\  P  e.  A )  ->  ( q  .<  (
q ( join `  K
) P )  <->  q  =/=  P ) )
2210, 12, 11, 21syl3anc 1228 . . . . . . . 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 232 . . . . . . 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 34178 . . . . . . . . 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 34104 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
28273ad2ant2 1018 . . . . . . . 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 15546 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  =  ( q (
join `  K ) P ) )
3025, 26, 28, 29syl3anc 1228 . . . . . . 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 4473 . . . . . 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 1025 . . . . . 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 34180 . . . . . . . 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 15538 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  e.  B )
3625, 26, 28, 35syl3anc 1228 . . . . . . 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 1091 . . . . . . 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 15458 . . . . . . 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 1230 . . . . . 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 679 . . . . 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 532 . . . 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 1195 . . 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 2935 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   Posetcpo 15427   ltcplt 15428   joincjn 15431   Latclat 15532    <o ccvr 34077   Atomscatm 34078   HLchlt 34165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166
This theorem is referenced by:  cdlemb  34608  lhpexle1  34822
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