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Theorem cdlemg18a 30792
Description: Show two lines are different. TODO: fix comment. (Contributed by NM, 14-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg18a  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
) ) )  -> 
( P  .\/  ( F `  Q )
)  =/=  ( Q 
.\/  ( F `  P ) ) )

Proof of Theorem cdlemg18a
StepHypRef Expression
1 simp3r 986 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
) ) )  -> 
( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) )
2 simpl1l 1008 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  K  e.  HL )
3 simpl21 1035 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  P  e.  A )
4 simpl1 960 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
5 simpl23 1037 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  F  e.  T )
6 simpl22 1036 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  Q  e.  A )
7 cdlemg12.l . . . . . . . 8  |-  .<_  =  ( le `  K )
8 cdlemg12.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
9 cdlemg12.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
10 cdlemg12.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
117, 8, 9, 10ltrnat 30254 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
124, 5, 6, 11syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( F `  Q )  e.  A )
137, 8, 9, 10ltrnat 30254 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
144, 5, 3, 13syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( F `  P )  e.  A )
15 simpl3l 1012 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  P  =/=  Q )
168, 9, 10ltrn11at 30261 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
) )  ->  ( F `  P )  =/=  ( F `  Q
) )
174, 5, 3, 6, 15, 16syl113anc 1196 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( F `  P )  =/=  ( F `  Q
) )
1817necomd 2633 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( F `  Q )  =/=  ( F `  P
) )
19 simpr 448 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( P  .\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )
20 cdlemg12.j . . . . . . 7  |-  .\/  =  ( join `  K )
2120, 8hlatexch4 29595 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  ( F `  Q )  e.  A )  /\  ( Q  e.  A  /\  ( F `  P
)  e.  A )  /\  ( P  =/= 
Q  /\  ( F `  Q )  =/=  ( F `  P )  /\  ( P  .\/  ( F `  Q )
)  =  ( Q 
.\/  ( F `  P ) ) ) )  ->  ( P  .\/  Q )  =  ( ( F `  Q
)  .\/  ( F `  P ) ) )
222, 3, 12, 6, 14, 15, 18, 19, 21syl323anc 1214 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  ( P  .\/  Q )  =  ( ( F `  Q )  .\/  ( F `  P )
) )
2322eqcomd 2392 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  ( ( F `  Q )  .\/  ( F `  P )
)  =/=  ( P 
.\/  Q ) ) )  /\  ( P 
.\/  ( F `  Q ) )  =  ( Q  .\/  ( F `  P )
) )  ->  (
( F `  Q
)  .\/  ( F `  P ) )  =  ( P  .\/  Q
) )
2423ex 424 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  ( F `  Q ) )  =  ( Q 
.\/  ( F `  P ) )  -> 
( ( F `  Q )  .\/  ( F `  P )
)  =  ( P 
.\/  Q ) ) )
2524necon3d 2588 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
) ) )  -> 
( ( ( F `
 Q )  .\/  ( F `  P ) )  =/=  ( P 
.\/  Q )  -> 
( P  .\/  ( F `  Q )
)  =/=  ( Q 
.\/  ( F `  P ) ) ) )
261, 25mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  F  e.  T )  /\  ( P  =/=  Q  /\  (
( F `  Q
)  .\/  ( F `  P ) )  =/=  ( P  .\/  Q
) ) )  -> 
( P  .\/  ( F `  Q )
)  =/=  ( Q 
.\/  ( F `  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   ` cfv 5394  (class class class)co 6020   lecple 13463   joincjn 14328   meetcmee 14329   Atomscatm 29378   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   trLctrl 30272
This theorem is referenced by:  cdlemg18c  30794
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-p0 14395  df-lat 14402  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219
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