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Theorem 2atlt 29628
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 29479 . . 3  |-  ( P  e.  A  ->  P  e.  B )
4 eqid 2283 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 2atomslt.s . . . 4  |-  .<  =  ( lt `  K )
6 eqid 2283 . . . 4  |-  ( join `  K )  =  (
join `  K )
71, 4, 5, 6, 2hlrelat 29591 . . 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 1231 . 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 983 . . . . . . . 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 1048 . . . . . . . . 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 1049 . . . . . . . . 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 956 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  (  <o  `  K )  =  ( 
<o  `  K )
145, 6, 2, 13atltcvr 29624 . . . . . . . . 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 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 ) )  -> 
( P  .<  ( P ( join `  K
) q )  <->  P (  <o  `  K ) ( P ( join `  K
) q ) ) )
169, 15mpbid 201 . . . . . . 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 29606 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  =/=  q  <->  P (  <o  `  K )
( P ( join `  K ) q ) ) )
1810, 11, 12, 17syl3anc 1182 . . . . . . 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 223 . . . . . 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 2529 . . . . 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 29626 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  q  e.  A  /\  P  e.  A )  ->  ( q  .<  (
q ( join `  K
) P )  <->  q  =/=  P ) )
2210, 12, 11, 21syl3anc 1182 . . . . . . . 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 223 . . . . . . 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 29553 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
2510, 24syl 15 . . . . . . . 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 15 . . . . . . . 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 29479 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
28273ad2ant2 977 . . . . . . . 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 14165 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  =  ( q (
join `  K ) P ) )
3025, 26, 28, 29syl3anc 1182 . . . . . . 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 4049 . . . . . 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 984 . . . . . 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 29555 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Poset )
3410, 33syl 15 . . . . . . 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 14156 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  e.  B )
3625, 26, 28, 35syl3anc 1182 . . . . . . 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 1050 . . . . . . 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 14105 . . . . . . 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 1184 . . . . . 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 660 . . . . 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 518 . . . 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 1150 . . 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 2653 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075   joincjn 14078   Latclat 14151    <o ccvr 29452   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  cdlemb  29983  lhpexle1  30197
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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