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Theorem cvrat 33063
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 25788 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 33062 . . 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 33005 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 465 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
9 simpr2 995 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
101, 5atbase 32931 . . . . . . . . 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 996 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
131, 5atbase 32931 . . . . . . . . 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 15227 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
168, 11, 14, 15syl3anc 1218 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1716breq2d 4302 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  <->  X  .<  ( Q 
.\/  P ) ) )
1817anbi2d 703 . . . . 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 457 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
20 simpr1 994 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
211, 2, 3, 4, 5cvratlem 33062 . . . . . . 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 434 . . . . . 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 1220 . . . . 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 215 . . . 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 429 . . 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 33007 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Poset )
2726adantr 465 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Poset )
281, 3latjcl 15219 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
298, 11, 14, 28syl3anc 1218 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
30 eqid 2441 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
311, 30, 2pltnle 15134 . . . . . . . . 9  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X  .<  ( P  .\/  Q ) )  ->  -.  ( P  .\/  Q ) ( le `  K
) X )
3231ex 434 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
3327, 20, 29, 32syl3anc 1218 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
341, 30, 3latjle12 15230 . . . . . . . . 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 1220 . . . . . . . 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 207 . . . . . . 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 140 . . . . . 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 488 . . . . . 6  |-  ( -.  ( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( -.  P
( le `  K
) X  \/  -.  Q ( le `  K ) X ) )
3937, 38syl6ib 226 . . . . 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 429 . . . 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 715 . . 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 381 . 2  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  ->  X  e.  A )
4342ex 434 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Basecbs 14172   lecple 14243   Posetcpo 15108   ltcplt 15109   joincjn 15112   0.cp0 15205   Latclat 15213   Atomscatm 32905   HLchlt 32992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993
This theorem is referenced by:  cvrat2  33070
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