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Theorem iscvlat2N 30196
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 30195 . 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 732 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  K  e.  AtLat )
7 simplrl 738 . . . . . . . 8  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  p  e.  A )
8 simpr 449 . . . . . . . 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 30189 . . . . . . . 8  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  x  e.  B )  ->  ( -.  p  .<_  x  <->  ( p  ./\  x )  =  .0.  ) )
126, 7, 8, 11syl3anc 1185 . . . . . . 7  |-  ( ( ( K  e.  AtLat  /\  ( p  e.  A  /\  q  e.  A
) )  /\  x  e.  B )  ->  ( -.  p  .<_  x  <->  ( p  ./\  x )  =  .0.  ) )
1312anbi1d 687 . . . . . 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 310 . . . . 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 2723 . . . 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 2747 . . 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 620 . 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 242 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   0.cp0 14471   Atomscatm 30135   AtLatcal 30136   CvLatclc 30137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-glb 14437  df-meet 14439  df-p0 14473  df-lat 14480  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194
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