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Theorem dihmeetlem6 32121
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 739 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  Q  .<_  W )
2 simpl1l 1006 . . . . . 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 30175 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 15 . . . . 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 959 . . . . . 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 960 . . . . . 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 14173 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1182 . . . . 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 738 . . . . . 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 30101 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  B )
1411, 13syl 15 . . . . 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 1007 . . . . . 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 30809 . . . . . 6  |-  ( W  e.  H  ->  W  e.  B )
1815, 17syl 15 . . . . 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 14184 . . . . 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 1184 . . . 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 447 . . . 4  |-  ( ( ( X  ./\  Y
)  .<_  W  /\  Q  .<_  W )  ->  Q  .<_  W )
2422, 23syl6bir 220 . . 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 168 . 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 733 . . . 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 32120 . . . 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 1190 . . 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 4049 . 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 292 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  dihjatc1  32123  dihmeetlem10N  32128
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-iin 3924  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-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  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-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799
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