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Theorem dihmeetlem6 31426
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.)
Hypotheses
Ref Expression
dihmeetlem6.b  |-  B  =  ( Base `  K
)
dihmeetlem6.l  |-  .<_  =  ( le `  K )
dihmeetlem6.h  |-  H  =  ( LHyp `  K
)
dihmeetlem6.j  |-  .\/  =  ( join `  K )
dihmeetlem6.m  |-  ./\  =  ( meet `  K )
dihmeetlem6.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dihmeetlem6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  ( X  ./\  ( Y  .\/  Q ) )  .<_  W )

Proof of Theorem dihmeetlem6
StepHypRef Expression
1 simprlr 740 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  Q  .<_  W )
2 simpl1l 1008 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  K  e.  HL )
3 hllat 29480 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  K  e.  Lat )
5 simpl2 961 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  X  e.  B )
6 simpl3 962 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  Y  e.  B )
7 dihmeetlem6.b . . . . . . 7  |-  B  =  ( Base `  K
)
8 dihmeetlem6.m . . . . . . 7  |-  ./\  =  ( meet `  K )
97, 8latmcl 14409 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( X  ./\ 
Y )  e.  B
)
11 simprll 739 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  Q  e.  A )
12 dihmeetlem6.a . . . . . . 7  |-  A  =  ( Atoms `  K )
137, 12atbase 29406 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
1411, 13syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  Q  e.  B )
15 simpl1r 1009 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  W  e.  H )
16 dihmeetlem6.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
177, 16lhpbase 30114 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1815, 17syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  W  e.  B )
19 dihmeetlem6.l . . . . . 6  |-  .<_  =  ( le `  K )
20 dihmeetlem6.j . . . . . 6  |-  .\/  =  ( join `  K )
217, 19, 20latjle12 14420 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Y )  e.  B  /\  Q  e.  B  /\  W  e.  B )
)  ->  ( (
( X  ./\  Y
)  .<_  W  /\  Q  .<_  W )  <->  ( ( X  ./\  Y )  .\/  Q )  .<_  W )
)
224, 10, 14, 18, 21syl13anc 1186 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( (
( X  ./\  Y
)  .<_  W  /\  Q  .<_  W )  <->  ( ( X  ./\  Y )  .\/  Q )  .<_  W )
)
23 simpr 448 . . . 4  |-  ( ( ( X  ./\  Y
)  .<_  W  /\  Q  .<_  W )  ->  Q  .<_  W )
2422, 23syl6bir 221 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( (
( X  ./\  Y
)  .\/  Q )  .<_  W  ->  Q  .<_  W ) )
251, 24mtod 170 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  (
( X  ./\  Y
)  .\/  Q )  .<_  W )
26 simprr 734 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  Q  .<_  X )
277, 19, 20, 8, 12dihmeetlem5 31425 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Q  e.  A  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q
) )  =  ( ( X  ./\  Y
)  .\/  Q )
)
282, 5, 6, 11, 26, 27syl32anc 1192 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( X  ./\  ( Y  .\/  Q
) )  =  ( ( X  ./\  Y
)  .\/  Q )
)
2928breq1d 4165 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  ( ( X  ./\  ( Y  .\/  Q ) )  .<_  W  <->  ( ( X  ./\  Y )  .\/  Q )  .<_  W )
)
3025, 29mtbird 293 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  ( X  ./\  ( Y  .\/  Q ) )  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   meetcmee 14331   Latclat 14403   Atomscatm 29380   HLchlt 29467   LHypclh 30100
This theorem is referenced by:  dihjatc1  31428  dihmeetlem10N  31433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-lat 14404  df-clat 14466  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104
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