Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg18a Unicode version

Theorem cdlemg18a 30867
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 30329 . . . . . . 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 30329 . . . . . . 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 30336 . . . . . . . 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 2529 . . . . . 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 29670 . . . . . 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 2288 . . . 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 2484 . 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 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg18c  30869
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-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-p0 14145  df-lat 14152  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-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294
  Copyright terms: Public domain W3C validator