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Theorem iscvlat2N 29514
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 29513 . 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 730 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  K  e.  AtLat )
7 simplrl 736 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  p  e.  A )
8 simpr 447 . . . . . . . 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 29507 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  x  e.  B )  ->  ( -.  p  .<_  x  <->  ( p  ./\  x )  =  .0.  ) )
126, 7, 8, 11syl3anc 1182 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  ( -.  p  .<_  x  <->  ( p  ./\  x )  =  .0.  ) )
1312anbi1d 685 . . . . . 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 308 . . . . 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 2559 . . . 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 2583 . . 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 618 . 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 240 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Atomscatm 29453   AtLatcal 29454   CvLatclc 29455
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  df-cvlat 29512
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