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Theorem atnle 29507
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 22956 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 958 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  K  e.  AtLat )
2 atllat 29490 . . . . . . . . 9  |-  ( K  e.  AtLat  ->  K  e.  Lat )
323ad2ant1 976 . . . . . . . 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 29479 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
763ad2ant2 977 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  P  e.  B )
8 simp3 957 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  X  e.  B )
9 atnle.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
104, 9latmcl 14157 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  ./\  X
)  e.  B )
113, 7, 8, 10syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( P  ./\  X )  e.  B )
1211adantr 451 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  ( P  ./\  X
)  e.  B )
13 simpr 447 . . . . . 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 29506 . . . . . 6  |-  ( ( K  e.  AtLat  /\  ( P  ./\  X )  e.  B  /\  ( P 
./\  X )  =/= 
.0.  )  ->  E. y  e.  A  y  .<_  ( P  ./\  X )
)
171, 12, 13, 16syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  E. y  e.  A  y  .<_  ( P  ./\  X ) )
18 simpl1 958 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  K  e.  AtLat
)
1918, 2syl 15 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  K  e.  Lat )
204, 5atbase 29479 . . . . . . . . . 10  |-  ( y  e.  A  ->  y  e.  B )
2120adantl 452 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  y  e.  B )
22 simpl2 959 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  P  e.  A )
2322, 6syl 15 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  P  e.  B )
24 simpl3 960 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  X  e.  B )
254, 14, 9latlem12 14184 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( (
y  .<_  P  /\  y  .<_  X )  <->  y  .<_  ( P  ./\  X )
) )
27 simpr 447 . . . . . . . . . . 11  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  y  e.  A )
2814, 5atcmp 29501 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  y  e.  A  /\  P  e.  A )  ->  (
y  .<_  P  <->  y  =  P ) )
2918, 27, 22, 28syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( y  .<_  P  <->  y  =  P ) )
30 breq1 4026 . . . . . . . . . . 11  |-  ( y  =  P  ->  (
y  .<_  X  <->  P  .<_  X ) )
3130biimpd 198 . . . . . . . . . 10  |-  ( y  =  P  ->  (
y  .<_  X  ->  P  .<_  X ) )
3229, 31syl6bi 219 . . . . . . . . 9  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( y  .<_  P  ->  ( y  .<_  X  ->  P  .<_  X ) ) )
3332imp3a 420 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( (
y  .<_  P  /\  y  .<_  X )  ->  P  .<_  X ) )
3426, 33sylbird 226 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  y  e.  A
)  ->  ( y  .<_  ( P  ./\  X
)  ->  P  .<_  X ) )
3534adantlr 695 . . . . . 6  |-  ( ( ( ( K  e. 
AtLat  /\  P  e.  A  /\  X  e.  B
)  /\  ( P  ./\ 
X )  =/=  .0.  )  /\  y  e.  A
)  ->  ( y  .<_  ( P  ./\  X
)  ->  P  .<_  X ) )
3635rexlimdva 2667 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  ( E. y  e.  A  y  .<_  ( P 
./\  X )  ->  P  .<_  X ) )
3717, 36mpd 14 . . . 4  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =/=  .0.  )  ->  P  .<_  X )
3837ex 423 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  (
( P  ./\  X
)  =/=  .0.  ->  P 
.<_  X ) )
3938necon1bd 2514 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  -> 
( P  ./\  X
)  =  .0.  )
)
4015, 5atn0 29498 . . . 4  |-  ( ( K  e.  AtLat  /\  P  e.  A )  ->  P  =/=  .0.  )
41403adant3 975 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  P  =/=  .0.  )
424, 14, 9latleeqm1 14185 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .<_  X  <->  ( P  ./\ 
X )  =  P ) )
433, 7, 8, 42syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( P  .<_  X  <->  ( P  ./\ 
X )  =  P ) )
4443adantr 451 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  X  <->  ( P  ./\ 
X )  =  P ) )
45 eqeq1 2289 . . . . . . . 8  |-  ( ( P  ./\  X )  =  P  ->  ( ( P  ./\  X )  =  .0.  <->  P  =  .0.  ) )
4645biimpcd 215 . . . . . . 7  |-  ( ( P  ./\  X )  =  .0.  ->  ( ( P  ./\  X )  =  P  ->  P  =  .0.  ) )
4746adantl 452 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( ( P  ./\  X )  =  P  ->  P  =  .0.  )
)
4844, 47sylbid 206 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  .<_  X  ->  P  =  .0.  )
)
4948necon3ad 2482 . . . 4  |-  ( ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  /\  ( P  ./\  X
)  =  .0.  )  ->  ( P  =/=  .0.  ->  -.  P  .<_  X ) )
5049ex 423 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  (
( P  ./\  X
)  =  .0.  ->  ( P  =/=  .0.  ->  -.  P  .<_  X )
) )
5141, 50mpid 37 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  (
( P  ./\  X
)  =  .0.  ->  -.  P  .<_  X )
)
5239, 51impbid 183 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 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   meetcmee 14079   0.cp0 14143   Latclat 14151   Atomscatm 29453   AtLatcal 29454
This theorem is referenced by:  atnem0  29508  iscvlat2N  29514  cvlexch3  29522  cvlexch4N  29523  cvlcvrp  29530  intnatN  29596  cvrat4  29632  dalem24  29886  cdlema2N  29981  llnexchb2lem  30057  lhpmat  30219  ltrnmw  30340  cdleme15b  30464  cdlemednpq  30488  cdleme20zN  30490  cdleme20y  30491  cdleme22cN  30531  dihmeetlem7N  31500  dihmeetlem17N  31513
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-glb 14109  df-meet 14111  df-p0 14145  df-lat 14152  df-covers 29456  df-ats 29457  df-atl 29488
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