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Theorem cdlemg17dALTN 31475
Description: Same as cdlemg17dN 31474 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 983 . . 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 985 . . . 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 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 ) )  ->  W  e.  H )
4 simp13 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 ) )  ->  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 30995 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
102, 3, 4, 9syl21anc 1181 . . 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 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
122, 11syl 15 . . . 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 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 6, 7, 8trlcl 30975 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
152, 3, 4, 14syl21anc 1181 . . . 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 1072 . . . . 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 989 . . . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
212, 16, 17, 20syl3anc 1182 . . . 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 30809 . . . . 5  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
233, 22syl 15 . . . 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 14200 . . . 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 1184 . . 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 887 . 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 30172 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
292, 28syl 15 . . 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 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 ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
31 simp3r 984 . . . 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 30980 . . . 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 1192 . . 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 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  =/=  Q )
355, 18, 24, 19, 6lhpat 30854 . . . 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 1192 . . 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 30123 . . 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 1182 . 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 201 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   AtLatcal 30076   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   trLctrl 30969
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-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  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  df-trl 30970
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