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Theorem cdlemg18a 31489
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 984 . 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 1006 . . . . . 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 1033 . . . . . 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 958 . . . . . . 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 1035 . . . . . . 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 1034 . . . . . . 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 30951 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
124, 5, 6, 11syl3anc 1182 . . . . . 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 30951 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
144, 5, 3, 13syl3anc 1182 . . . . . 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 1010 . . . . . 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 30958 . . . . . . . 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 1194 . . . . . . 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 2542 . . . . . 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 447 . . . . . 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 30292 . . . . . 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 1212 . . . . 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 2301 . . . 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 423 . . 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 2497 . 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 14 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   meetcmee 14095   Atomscatm 30075   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
This theorem is referenced by:  cdlemg18c  31491
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-map 6790  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-p0 14161  df-lat 14168  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916
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