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Theorem iscvlat2N 30136
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 30135 . 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 30129 . . . . . . . 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 2572 . . . 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 2596 . . 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 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   0.cp0 14159   Atomscatm 30075   AtLatcal 30076   CvLatclc 30077
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  df-cvlat 30134
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