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Theorem lplnllnneN 30427
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 2438 . . 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 30421 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  Y  e.  P )  ->  -.  Q ( le `  K ) ( R 
.\/  S ) )
7 simpl1 961 . . . . . 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 1036 . . . . . 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 1038 . . . . . 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 30246 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  S  e.  A )  ->  Q ( le `  K ) ( Q 
.\/  S ) )
117, 8, 9, 10syl3anc 1185 . . . . 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 449 . . . . 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 4239 . . . 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 425 . . 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 2640 . 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   lecple 13541   joincjn 14406   Atomscatm 30135   HLchlt 30222   LPlanesclpl 30363
This theorem is referenced by:  cdleme16aN  31130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370
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