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Theorem cdlemg17dALTN 31398
Description: Same as cdlemg17dN 31397 with fewer antecedents but longer proof TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)
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
cdlemg17dALTN  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  =  ( ( P  .\/  Q ) 
./\  W ) )

Proof of Theorem cdlemg17dALTN
StepHypRef Expression
1 simp3l 985 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  .<_  ( P  .\/  Q ) )
2 simp11 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  K  e.  HL )
3 simp12 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  W  e.  H )
4 simp13 989 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  G  e.  T )
5 cdlemg12.l . . . . 5  |-  .<_  =  ( le `  K )
6 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
8 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
95, 6, 7, 8trlle 30918 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
102, 3, 4, 9syl21anc 1183 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  .<_  W )
11 hllat 30098 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
122, 11syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  K  e.  Lat )
13 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 6, 7, 8trlcl 30898 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
152, 3, 4, 14syl21anc 1183 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  e.  ( Base `  K ) )
16 simp21l 1074 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  P  e.  A )
17 simp22 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  Q  e.  A )
18 cdlemg12.j . . . . . 6  |-  .\/  =  ( join `  K )
19 cdlemg12.a . . . . . 6  |-  A  =  ( Atoms `  K )
2013, 18, 19hlatjcl 30101 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
212, 16, 17, 20syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
2213, 6lhpbase 30732 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
233, 22syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  W  e.  ( Base `  K ) )
24 cdlemg12.m . . . . 5  |-  ./\  =  ( meet `  K )
2513, 5, 24latlem12 14499 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( R `  G )  e.  (
Base `  K )  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( ( R `
 G )  .<_  ( P  .\/  Q )  /\  ( R `  G )  .<_  W )  <-> 
( R `  G
)  .<_  ( ( P 
.\/  Q )  ./\  W ) ) )
2612, 15, 21, 23, 25syl13anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( ( ( R `
 G )  .<_  ( P  .\/  Q )  /\  ( R `  G )  .<_  W )  <-> 
( R `  G
)  .<_  ( ( P 
.\/  Q )  ./\  W ) ) )
271, 10, 26mpbi2and 888 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  .<_  ( ( P 
.\/  Q )  ./\  W ) )
28 hlatl 30095 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
292, 28syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  K  e.  AtLat )
30 simp21 990 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
31 simp3r 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( G `  P
)  =/=  P )
325, 19, 6, 7, 8trlat 30903 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( G  e.  T  /\  ( G `  P )  =/=  P ) )  ->  ( R `  G )  e.  A
)
332, 3, 30, 4, 31, 32syl212anc 1194 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  e.  A )
34 simp23 992 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  ->  P  =/=  Q )
355, 18, 24, 19, 6lhpat 30777 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  ( ( P 
.\/  Q )  ./\  W )  e.  A )
362, 3, 30, 17, 34, 35syl212anc 1194 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( ( P  .\/  Q )  ./\  W )  e.  A )
375, 19atcmp 30046 . . 3  |-  ( ( K  e.  AtLat  /\  ( R `  G )  e.  A  /\  (
( P  .\/  Q
)  ./\  W )  e.  A )  ->  (
( R `  G
)  .<_  ( ( P 
.\/  Q )  ./\  W )  <->  ( R `  G )  =  ( ( P  .\/  Q
)  ./\  W )
) )
3829, 33, 36, 37syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( ( R `  G )  .<_  ( ( P  .\/  Q ) 
./\  W )  <->  ( R `  G )  =  ( ( P  .\/  Q
)  ./\  W )
) )
3927, 38mpbid 202 1  |-  ( ( ( K  e.  HL  /\  W  e.  H  /\  G  e.  T )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  P  =/= 
Q )  /\  (
( R `  G
)  .<_  ( P  .\/  Q )  /\  ( G `
 P )  =/= 
P ) )  -> 
( R `  G
)  =  ( ( P  .\/  Q ) 
./\  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   AtLatcal 29999   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   trLctrl 30892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-map 7012  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893
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