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Theorem cvrat 29611
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22966 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 29610 . . 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 29553 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
87adantr 451 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
9 simpr2 962 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
101, 5atbase 29479 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  B )
119, 10syl 15 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
12 simpr3 963 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
131, 5atbase 29479 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  B )
1412, 13syl 15 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
151, 3latjcom 14165 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
168, 11, 14, 15syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  =  ( Q  .\/  P
) )
1716breq2d 4035 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  <->  X  .<  ( Q 
.\/  P ) ) )
1817anbi2d 684 . . . . 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 443 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
20 simpr1 961 . . . . . 6  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
211, 2, 3, 4, 5cvratlem 29610 . . . . . . 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 423 . . . . . 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 1184 . . . . 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 206 . . . 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 418 . . 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 29555 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Poset )
2726adantr 451 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Poset )
281, 3latjcl 14156 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
298, 11, 14, 28syl3anc 1182 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
30 eqid 2283 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
311, 30, 2pltnle 14100 . . . . . . . . 9  |-  ( ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  /\  X  .<  ( P  .\/  Q ) )  ->  -.  ( P  .\/  Q ) ( le `  K
) X )
3231ex 423 . . . . . . . 8  |-  ( ( K  e.  Poset  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
3327, 20, 29, 32syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<  ( P  .\/  Q )  ->  -.  ( P  .\/  Q ) ( le `  K ) X ) )
341, 30, 3latjle12 14168 . . . . . . . . 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 1184 . . . . . . . 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 198 . . . . . . 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 132 . . . . . 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 474 . . . . . 6  |-  ( -.  ( P ( le
`  K ) X  /\  Q ( le
`  K ) X )  <->  ( -.  P
( le `  K
) X  \/  -.  Q ( le `  K ) X ) )
3937, 38syl6ib 217 . . . . 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 418 . . . 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 696 . . 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 370 . 2  |-  ( ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  /\  ( X  =/=  .0.  /\  X  .<  ( P  .\/  Q
) ) )  ->  X  e.  A )
4342ex 423 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 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   Posetcpo 14074   ltcplt 14075   joincjn 14078   0.cp0 14143   Latclat 14151   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  cvrat2  29618
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-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541
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