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Theorem cdlemg17dALTN 36787
Description: Same as cdlemg17dN 36786 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 1022 . . 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 1024 . . . 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 1025 . . . 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 1026 . . . 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 36306 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  .<_  W )
102, 3, 4, 9syl21anc 1225 . . 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 35485 . . . . 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 2454 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 6, 7, 8trlcl 36286 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T
)  ->  ( R `  G )  e.  (
Base `  K )
)
152, 3, 4, 14syl21anc 1225 . . . 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 1111 . . . . 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 1028 . . . . 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 35488 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
212, 16, 17, 20syl3anc 1226 . . . 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 36119 . . . . 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 15907 . . . 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 1228 . . 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 919 . 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 35482 . . . 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 1027 . . . 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 1023 . . . 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 36291 . . . 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 1236 . . 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 1029 . . . 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 36164 . . . 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 1236 . . 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 35433 . . 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 1226 . 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 210 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   meetcmee 15773   Latclat 15874   Atomscatm 35385   AtLatcal 35386   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   trLctrl 36280
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 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281
This theorem is referenced by: (None)
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