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Theorem iscvlat2N 29819
Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
iscvlat2.b  |-  B  =  ( Base `  K
)
iscvlat2.l  |-  .<_  =  ( le `  K )
iscvlat2.j  |-  .\/  =  ( join `  K )
iscvlat2.m  |-  ./\  =  ( meet `  K )
iscvlat2.z  |-  .0.  =  ( 0. `  K )
iscvlat2.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
iscvlat2N  |-  ( K  e.  CvLat 
<->  ( K  e.  AtLat  /\ 
A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( ( p 
./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
Distinct variable groups:    q, p, x, A    x, B    K, p, q, x
Allowed substitution hints:    B( q, p)    .\/ ( x, q, p)    .<_ ( x, q, p)    ./\ ( x, q, p)    .0. ( x, q, p)

Proof of Theorem iscvlat2N
StepHypRef Expression
1 iscvlat2.b . . 3  |-  B  =  ( Base `  K
)
2 iscvlat2.l . . 3  |-  .<_  =  ( le `  K )
3 iscvlat2.j . . 3  |-  .\/  =  ( join `  K )
4 iscvlat2.a . . 3  |-  A  =  ( Atoms `  K )
51, 2, 3, 4iscvlat 29818 . 2  |-  ( K  e.  CvLat 
<->  ( K  e.  AtLat  /\ 
A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
6 simpll 731 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  K  e.  AtLat )
7 simplrl 737 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  p  e.  A )
8 simpr 448 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  x  e.  B )
9 iscvlat2.m . . . . . . . . 9  |-  ./\  =  ( meet `  K )
10 iscvlat2.z . . . . . . . . 9  |-  .0.  =  ( 0. `  K )
111, 2, 9, 10, 4atnle 29812 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  x  e.  B )  ->  ( -.  p  .<_  x  <->  ( p  ./\  x )  =  .0.  ) )
126, 7, 8, 11syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  ( -.  p  .<_  x  <->  ( p  ./\  x )  =  .0.  ) )
1312anbi1d 686 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  (
( -.  p  .<_  x  /\  p  .<_  ( x 
.\/  q ) )  <-> 
( ( p  ./\  x )  =  .0. 
/\  p  .<_  ( x 
.\/  q ) ) ) )
1413imbi1d 309 . . . . 5  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  (
( ( -.  p  .<_  x  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) )  <->  ( ( ( p  ./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
1514ralbidva 2690 . . . 4  |-  ( ( K  e.  AtLat  /\  (
p  e.  A  /\  q  e.  A )
)  ->  ( A. x  e.  B  (
( -.  p  .<_  x  /\  p  .<_  ( x 
.\/  q ) )  ->  q  .<_  ( x 
.\/  p ) )  <->  A. x  e.  B  ( ( ( p 
./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
16152ralbidva 2714 . . 3  |-  ( K  e.  AtLat  ->  ( A. p  e.  A  A. q  e.  A  A. x  e.  B  (
( -.  p  .<_  x  /\  p  .<_  ( x 
.\/  q ) )  ->  q  .<_  ( x 
.\/  p ) )  <->  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( ( p 
./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
1716pm5.32i 619 . 2  |-  ( ( K  e.  AtLat  /\  A. p  e.  A  A. q  e.  A  A. x  e.  B  (
( -.  p  .<_  x  /\  p  .<_  ( x 
.\/  q ) )  ->  q  .<_  ( x 
.\/  p ) ) )  <->  ( K  e. 
AtLat  /\  A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( ( p  ./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
185, 17bitri 241 1  |-  ( K  e.  CvLat 
<->  ( K  e.  AtLat  /\ 
A. p  e.  A  A. q  e.  A  A. x  e.  B  ( ( ( p 
./\  x )  =  .0.  /\  p  .<_  ( x  .\/  q ) )  ->  q  .<_  ( x  .\/  p ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   0.cp0 14429   Atomscatm 29758   AtLatcal 29759   CvLatclc 29760
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-glb 14395  df-meet 14397  df-p0 14431  df-lat 14438  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817
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