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Theorem cdlemg18a 36820
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 1023 . 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 1045 . . . . . 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 1072 . . . . . 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 997 . . . . . . 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 1074 . . . . . . 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 1073 . . . . . . 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 36280 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  Q  e.  A
)  ->  ( F `  Q )  e.  A
)
124, 5, 6, 11syl3anc 1226 . . . . . 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 36280 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  A
)  ->  ( F `  P )  e.  A
)
144, 5, 3, 13syl3anc 1226 . . . . . 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 1049 . . . . . 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 36287 . . . . . . . 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 1238 . . . . . . 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 2725 . . . . . 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 459 . . . . . 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 35621 . . . . . 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 1256 . . . . 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 2462 . . . 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 432 . . 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 2678 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   ` cfv 5570  (class class class)co 6270   lecple 14794   joincjn 15775   meetcmee 15776   Atomscatm 35404   HLchlt 35491   LHypclh 36124   LTrncltrn 36241   trLctrl 36299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-preset 15759  df-poset 15777  df-plt 15790  df-lub 15806  df-glb 15807  df-join 15808  df-meet 15809  df-p0 15871  df-lat 15878  df-oposet 35317  df-ol 35319  df-oml 35320  df-covers 35407  df-ats 35408  df-atl 35439  df-cvlat 35463  df-hlat 35492  df-lhyp 36128  df-laut 36129  df-ldil 36244  df-ltrn 36245
This theorem is referenced by:  cdlemg18c  36822
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