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Theorem cdlemg18a 34641
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 1017 . 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 1039 . . . . . 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 1066 . . . . . 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 991 . . . . . . 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 1068 . . . . . . 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 1067 . . . . . . 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 34103 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
124, 5, 6, 11syl3anc 1219 . . . . . 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 34103 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
144, 5, 3, 13syl3anc 1219 . . . . . 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 1043 . . . . . 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 34110 . . . . . . . 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 1231 . . . . . . 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 2720 . . . . . 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 461 . . . . . 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 33444 . . . . . 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 1249 . . . . 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 2460 . . . 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 434 . . 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 2673 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   ` cfv 5521  (class class class)co 6195   lecple 14359   joincjn 15228   meetcmee 15229   Atomscatm 33227   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   trLctrl 34121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-lat 15330  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068
This theorem is referenced by:  cdlemg18c  34643
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