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Theorem cvrat 30293
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 23894 analog.) (Contributed by NM, 22-Nov-2011.)
Hypotheses
Ref Expression
cvrat.b  |-  B  =  ( Base `  K
)
cvrat.s  |-  .<  =  ( lt `  K )
cvrat.j  |-  .\/  =  ( join `  K )
cvrat.z  |-  .0.  =  ( 0. `  K )
cvrat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvrat  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  ->  X  e.  A )
)

Proof of Theorem cvrat
StepHypRef Expression
1 cvrat.b . . . 4  |-  B  =  ( Base `  K
)
2 cvrat.s . . . 4  |-  .<  =  ( lt `  K )
3 cvrat.j . . . 4  |-  .\/  =  ( join `  K )
4 cvrat.z . . . 4  |-  .0.  =  ( 0. `  K )
5 cvrat.a . . . 4  |-  A  =  ( Atoms `  K )
61, 2, 3, 4, 5cvratlem 30292 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  -> 
( -.  P ( le `  K ) X  ->  X  e.  A ) )
7 hllat 30235 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 453 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
9 simpr2 965 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
101, 5atbase 30161 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
119, 10syl 16 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
12 simpr3 966 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
131, 5atbase 30161 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
1412, 13syl 16 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
151, 3latjcom 14493 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
168, 11, 14, 15syl3anc 1185 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1716breq2d 4227 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  <->  X  .<  ( Q 
.\/  P ) ) )
1817anbi2d 686 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  <->  ( X  =/=  .0.  /\  X  .<  ( Q  .\/  P ) ) ) )
19 simpl 445 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
20 simpr1 964 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
211, 2, 3, 4, 5cvratlem 30292 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( Q  .\/  P
) ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) )
2221ex 425 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  Q  e.  A  /\  P  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( Q  .\/  P ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) ) )
2319, 20, 12, 9, 22syl13anc 1187 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( Q  .\/  P ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) ) )
2418, 23sylbid 208 . . . 4  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) ) )
2524imp 420 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  -> 
( -.  Q ( le `  K ) X  ->  X  e.  A ) )
26 hlpos 30237 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Poset )
2726adantr 453 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Poset )
281, 3latjcl 14484 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
298, 11, 14, 28syl3anc 1185 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
30 eqid 2438 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
311, 30, 2pltnle 14428 . . . . . . . . 9  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X  .<  ( P  .\/  Q ) )  ->  -.  ( P  .\/  Q ) ( le `  K
) X )
3231ex 425 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
3327, 20, 29, 32syl3anc 1185 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
341, 30, 3latjle12 14496 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  B  /\  Q  e.  B  /\  X  e.  B
) )  ->  (
( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( P  .\/  Q ) ( le `  K ) X ) )
358, 11, 14, 20, 34syl13anc 1187 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( P  .\/  Q ) ( le `  K ) X ) )
3635biimpd 200 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  ->  ( P  .\/  Q ) ( le
`  K ) X ) )
3733, 36nsyld 135 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P ( le `  K ) X  /\  Q ( le `  K ) X ) ) )
38 ianor 476 . . . . . 6  |-  ( -.  ( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( -.  P
( le `  K
) X  \/  -.  Q ( le `  K ) X ) )
3937, 38syl6ib 219 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  ( -.  P ( le `  K ) X  \/  -.  Q ( le `  K ) X ) ) )
4039imp 420 . . . 4  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  X  .<  ( P  .\/  Q
) )  ->  ( -.  P ( le `  K ) X  \/  -.  Q ( le `  K ) X ) )
4140adantrl 698 . . 3  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  -> 
( -.  P ( le `  K ) X  \/  -.  Q
( le `  K
) X ) )
426, 25, 41mpjaod 372 . 2  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  ->  X  e.  A )
4342ex 425 1  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( X  =/=  .0.  /\  X  .<  ( P  .\/  Q ) )  ->  X  e.  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   Posetcpo 14402   ltcplt 14403   joincjn 14406   0.cp0 14471   Latclat 14479   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  cvrat2  30300
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-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223
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