Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2atlt Structured version   Unicode version

Theorem 2atlt 32742
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 32593 . . 3  |-  ( P  e.  A  ->  P  e.  B )
4 eqid 2420 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
5 2atomslt.s . . . 4  |-  .<  =  ( lt `  K )
6 eqid 2420 . . . 4  |-  ( join `  K )  =  (
join `  K )
71, 4, 5, 6, 2hlrelat 32705 . . 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 1313 . 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 1033 . . . . . . . 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 1098 . . . . . . . . 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 1099 . . . . . . . . 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 1006 . . . . . . . . 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 2420 . . . . . . . . . 10  |-  (  <o  `  K )  =  ( 
<o  `  K )
145, 6, 2, 13atltcvr 32738 . . . . . . . . 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 1266 . . . . . . . 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 213 . . . . . . 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 32720 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  q  e.  A )  ->  ( P  =/=  q  <->  P (  <o  `  K )
( P ( join `  K ) q ) ) )
1810, 11, 12, 17syl3anc 1264 . . . . . . 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 235 . . . . . 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 2693 . . . . 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 32740 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  q  e.  A  /\  P  e.  A )  ->  ( q  .<  (
q ( join `  K
) P )  <->  q  =/=  P ) )
2210, 12, 11, 21syl3anc 1264 . . . . . . . 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 235 . . . . . . 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 32667 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
2510, 24syl 17 . . . . . . . 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 17 . . . . . . . 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 32593 . . . . . . . . 9  |-  ( q  e.  A  ->  q  e.  B )
28273ad2ant2 1027 . . . . . . . 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 16249 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  =  ( q (
join `  K ) P ) )
3025, 26, 28, 29syl3anc 1264 . . . . . . 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 4443 . . . . . 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 1034 . . . . . 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 32669 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Poset )
3410, 33syl 17 . . . . . . 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 16241 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  q  e.  B )  ->  ( P ( join `  K ) q )  e.  B )
3625, 26, 28, 35syl3anc 1264 . . . . . . 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 1100 . . . . . . 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 16161 . . . . . . 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 1266 . . . . . 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 683 . . . . 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 534 . . . 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 1204 . . 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 2895 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   E.wrex 2774   class class class wbr 4417   ` cfv 5592  (class class class)co 6296   Basecbs 15073   lecple 15149   Posetcpo 16129   ltcplt 16130   joincjn 16133   Latclat 16235    <o ccvr 32566   Atomscatm 32567   HLchlt 32654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 16117  df-poset 16135  df-plt 16148  df-lub 16164  df-glb 16165  df-join 16166  df-meet 16167  df-p0 16229  df-lat 16236  df-clat 16298  df-oposet 32480  df-ol 32482  df-oml 32483  df-covers 32570  df-ats 32571  df-atl 32602  df-cvlat 32626  df-hlat 32655
This theorem is referenced by:  cdlemb  33097  lhpexle1  33311
  Copyright terms: Public domain W3C validator