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Theorem cvrat 30233
Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 22982 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 30232 . . 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 30175 . . . . . . . . 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 30101 . . . . . . . . 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 30101 . . . . . . . . 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 14181 . . . . . . . 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 4051 . . . . . 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 30232 . . . . . . 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 30177 . . . . . . . . 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 14172 . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
311, 30, 2pltnle 14116 . . . . . . . . 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 14184 . . . . . . . . 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   Posetcpo 14090   ltcplt 14091   joincjn 14094   0.cp0 14159   Latclat 14167   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  cvrat2  30240
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-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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