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Theorem lplnribN 29740
Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
islpln2a.l  |-  .<_  =  ( le `  K )
islpln2a.j  |-  .\/  =  ( join `  K )
islpln2a.a  |-  A  =  ( Atoms `  K )
islpln2a.p  |-  P  =  ( LPlanes `  K )
islpln2a.y  |-  Y  =  ( ( Q  .\/  R )  .\/  S )
Assertion
Ref Expression
lplnribN  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  -.  R  .<_  ( Q  .\/  S ) )

Proof of Theorem lplnribN
StepHypRef Expression
1 islpln2a.l . . . . . 6  |-  .<_  =  ( le `  K )
2 islpln2a.j . . . . . 6  |-  .\/  =  ( join `  K )
3 islpln2a.a . . . . . 6  |-  A  =  ( Atoms `  K )
41, 2, 33noncolr1N 29639 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( S  =/=  Q  /\  -.  R  .<_  ( S 
.\/  Q ) ) )
54simprd 449 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  R  .<_  ( S 
.\/  Q ) )
653expia 1153 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R ) )  ->  -.  R  .<_  ( S  .\/  Q ) ) )
7 islpln2a.p . . . 4  |-  P  =  ( LPlanes `  K )
8 islpln2a.y . . . 4  |-  Y  =  ( ( Q  .\/  R )  .\/  S )
91, 2, 3, 7, 8islpln2ah 29738 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Y  e.  P  <->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) ) )
102, 3hlatjcom 29557 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  ( Q  .\/  S
)  =  ( S 
.\/  Q ) )
11103adant3r2 1161 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  .\/  S )  =  ( S  .\/  Q
) )
1211breq2d 4035 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( R  .<_  ( Q  .\/  S )  <->  R  .<_  ( S 
.\/  Q ) ) )
1312notbid 285 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( -.  R  .<_  ( Q 
.\/  S )  <->  -.  R  .<_  ( S  .\/  Q
) ) )
146, 9, 133imtr4d 259 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Y  e.  P  ->  -.  R  .<_  ( Q  .\/  S ) ) )
15143impia 1148 1  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  -.  R  .<_  ( Q  .\/  S ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  lplnri3N  29744
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  df-llines 29687  df-lplanes 29688
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