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Theorem lplnllnneN 30038
Description: Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnri1.j  |-  .\/  =  ( join `  K )
lplnri1.a  |-  A  =  ( Atoms `  K )
lplnri1.p  |-  P  =  ( LPlanes `  K )
lplnri1.y  |-  Y  =  ( ( Q  .\/  R )  .\/  S )
Assertion
Ref Expression
lplnllnneN  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  ( Q  .\/  S )  =/=  ( R  .\/  S
) )

Proof of Theorem lplnllnneN
StepHypRef Expression
1 eqid 2404 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 lplnri1.j . . 3  |-  .\/  =  ( join `  K )
3 lplnri1.a . . 3  |-  A  =  ( Atoms `  K )
4 lplnri1.p . . 3  |-  P  =  ( LPlanes `  K )
5 lplnri1.y . . 3  |-  Y  =  ( ( Q  .\/  R )  .\/  S )
61, 2, 3, 4, 5lplnriaN 30032 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  -.  Q ( le `  K ) ( R 
.\/  S ) )
7 simpl1 960 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  /\  ( Q  .\/  S )  =  ( R  .\/  S
) )  ->  K  e.  HL )
8 simpl21 1035 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  /\  ( Q  .\/  S )  =  ( R  .\/  S
) )  ->  Q  e.  A )
9 simpl23 1037 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  /\  ( Q  .\/  S )  =  ( R  .\/  S
) )  ->  S  e.  A )
101, 2, 3hlatlej1 29857 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  Q ( le `  K ) ( Q 
.\/  S ) )
117, 8, 9, 10syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  /\  ( Q  .\/  S )  =  ( R  .\/  S
) )  ->  Q
( le `  K
) ( Q  .\/  S ) )
12 simpr 448 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  /\  ( Q  .\/  S )  =  ( R  .\/  S
) )  ->  ( Q  .\/  S )  =  ( R  .\/  S
) )
1311, 12breqtrd 4196 . . . 4  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  /\  ( Q  .\/  S )  =  ( R  .\/  S
) )  ->  Q
( le `  K
) ( R  .\/  S ) )
1413ex 424 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  (
( Q  .\/  S
)  =  ( R 
.\/  S )  ->  Q ( le `  K ) ( R 
.\/  S ) ) )
1514necon3bd 2604 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  ( -.  Q ( le `  K ) ( R 
.\/  S )  -> 
( Q  .\/  S
)  =/=  ( R 
.\/  S ) ) )
166, 15mpd 15 1  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  ( Q  .\/  S )  =/=  ( R  .\/  S
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   lecple 13491   joincjn 14356   Atomscatm 29746   HLchlt 29833   LPlanesclpl 29974
This theorem is referenced by:  cdleme16aN  30741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981
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