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Theorem atnle 30129
Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 22972 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 30112 . . . . . . . . 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 30101 . . . . . . . . 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 14173 . . . . . . . 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 30128 . . . . . 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 30101 . . . . . . . . . 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 14200 . . . . . . . . 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 30123 . . . . . . . . . . 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 4042 . . . . . . . . . . 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 2680 . . . . 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 2527 . 2  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P  .<_  X  -> 
( P  ./\  X
)  =  .0.  )
)
4015, 5atn0 30120 . . . 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 14201 . . . . . . . 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 2302 . . . . . . . 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 2495 . . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   meetcmee 14095   0.cp0 14159   Latclat 14167   Atomscatm 30075   AtLatcal 30076
This theorem is referenced by:  atnem0  30130  iscvlat2N  30136  cvlexch3  30144  cvlexch4N  30145  cvlcvrp  30152  intnatN  30218  cvrat4  30254  dalem24  30508  cdlema2N  30603  llnexchb2lem  30679  lhpmat  30841  ltrnmw  30962  cdleme15b  31086  cdlemednpq  31110  cdleme20zN  31112  cdleme20y  31113  cdleme22cN  31153  dihmeetlem7N  32122  dihmeetlem17N  32135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-glb 14125  df-meet 14127  df-p0 14161  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110
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