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Theorem atnle 29800
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 23832 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
atnle.b  |-  B  =  ( Base `  K
)
atnle.l  |-  .<_  =  ( le `  K )
atnle.m  |-  ./\  =  ( meet `  K )
atnle.z  |-  .0.  =  ( 0. `  K )
atnle.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atnle  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  <->  ( P  ./\ 
X )  =  .0.  ) )

Proof of Theorem atnle
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpl1 960 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  K  e.  AtLat )
2 atllat 29783 . . . . . . . . 9  |-  ( K  e.  AtLat  ->  K  e.  Lat )
323ad2ant1 978 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  K  e.  Lat )
4 atnle.b . . . . . . . . . 10  |-  B  =  ( Base `  K
)
5 atnle.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
64, 5atbase 29772 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
763ad2ant2 979 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  P  e.  B )
8 simp3 959 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  X  e.  B )
9 atnle.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
104, 9latmcl 14435 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  ./\  X
)  e.  B )
113, 7, 8, 10syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( P  ./\  X )  e.  B )
1211adantr 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  ( P  ./\  X
)  e.  B )
13 simpr 448 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  ( P  ./\  X
)  =/=  .0.  )
14 atnle.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 atnle.z . . . . . . 7  |-  .0.  =  ( 0. `  K )
164, 14, 15, 5atlex 29799 . . . . . 6  |-  ( ( K  e.  AtLat  /\  ( P  ./\  X )  e.  B  /\  ( P 
./\  X )  =/= 
.0.  )  ->  E. y  e.  A  y  .<_  ( P  ./\  X )
)
171, 12, 13, 16syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  E. y  e.  A  y  .<_  ( P  ./\  X ) )
18 simpl1 960 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  K  e.  AtLat
)
1918, 2syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  K  e.  Lat )
204, 5atbase 29772 . . . . . . . . . 10  |-  ( y  e.  A  ->  y  e.  B )
2120adantl 453 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  y  e.  B )
22 simpl2 961 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  P  e.  A )
2322, 6syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  P  e.  B )
24 simpl3 962 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  X  e.  B )
254, 14, 9latlem12 14462 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( y  e.  B  /\  P  e.  B  /\  X  e.  B
) )  ->  (
( y  .<_  P  /\  y  .<_  X )  <->  y  .<_  ( P  ./\  X )
) )
2619, 21, 23, 24, 25syl13anc 1186 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( (
y  .<_  P  /\  y  .<_  X )  <->  y  .<_  ( P  ./\  X )
) )
27 simpr 448 . . . . . . . . . . 11  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  y  e.  A )
2814, 5atcmp 29794 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  y  e.  A  /\  P  e.  A )  ->  (
y  .<_  P  <->  y  =  P ) )
2918, 27, 22, 28syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( y  .<_  P  <->  y  =  P ) )
30 breq1 4175 . . . . . . . . . . 11  |-  ( y  =  P  ->  (
y  .<_  X  <->  P  .<_  X ) )
3130biimpd 199 . . . . . . . . . 10  |-  ( y  =  P  ->  (
y  .<_  X  ->  P  .<_  X ) )
3229, 31syl6bi 220 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( y  .<_  P  ->  ( y  .<_  X  ->  P  .<_  X ) ) )
3332imp3a 421 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( (
y  .<_  P  /\  y  .<_  X )  ->  P  .<_  X ) )
3426, 33sylbird 227 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( y  .<_  ( P  ./\  X
)  ->  P  .<_  X ) )
3534adantlr 696 . . . . . 6  |-  ( ( ( ( K  e. 
AtLat  /\  P  e.  A  /\  X  e.  B
)  /\  ( P  ./\ 
X )  =/=  .0.  )  /\  y  e.  A
)  ->  ( y  .<_  ( P  ./\  X
)  ->  P  .<_  X ) )
3635rexlimdva 2790 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  ( E. y  e.  A  y  .<_  ( P 
./\  X )  ->  P  .<_  X ) )
3717, 36mpd 15 . . . 4  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  P  .<_  X )
3837ex 424 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  (
( P  ./\  X
)  =/=  .0.  ->  P 
.<_  X ) )
3938necon1bd 2635 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  -> 
( P  ./\  X
)  =  .0.  )
)
4015, 5atn0 29791 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
41403adant3 977 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  P  =/=  .0.  )
424, 14, 9latleeqm1 14463 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .<_  X  <->  ( P  ./\ 
X )  =  P ) )
433, 7, 8, 42syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( P  .<_  X  <->  ( P  ./\ 
X )  =  P ) )
4443adantr 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  X  <->  ( P  ./\ 
X )  =  P ) )
45 eqeq1 2410 . . . . . . . 8  |-  ( ( P  ./\  X )  =  P  ->  ( ( P  ./\  X )  =  .0.  <->  P  =  .0.  ) )
4645biimpcd 216 . . . . . . 7  |-  ( ( P  ./\  X )  =  .0.  ->  ( ( P  ./\  X )  =  P  ->  P  =  .0.  ) )
4746adantl 453 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( ( P  ./\  X )  =  P  ->  P  =  .0.  )
)
4844, 47sylbid 207 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  X  ->  P  =  .0.  )
)
4948necon3ad 2603 . . . 4  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  =/=  .0.  ->  -.  P  .<_  X ) )
5049ex 424 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  (
( P  ./\  X
)  =  .0.  ->  ( P  =/=  .0.  ->  -.  P  .<_  X )
) )
5141, 50mpid 39 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  (
( P  ./\  X
)  =  .0.  ->  -.  P  .<_  X )
)
5239, 51impbid 184 1  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  <->  ( P  ./\ 
X )  =  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> 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   meetcmee 14357   0.cp0 14421   Latclat 14429   Atomscatm 29746   AtLatcal 29747
This theorem is referenced by:  atnem0  29801  iscvlat2N  29807  cvlexch3  29815  cvlexch4N  29816  cvlcvrp  29823  intnatN  29889  cvrat4  29925  dalem24  30179  cdlema2N  30274  llnexchb2lem  30350  lhpmat  30512  ltrnmw  30633  cdleme15b  30757  cdlemednpq  30781  cdleme20zN  30783  cdleme20y  30784  cdleme22cN  30824  dihmeetlem7N  31793  dihmeetlem17N  31806
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-glb 14387  df-meet 14389  df-p0 14423  df-lat 14430  df-covers 29749  df-ats 29750  df-atl 29781
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