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Theorem dihmeetlem6 32181
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 741 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  B  /\  Y  e.  B )  /\  (
( Q  e.  A  /\  -.  Q  .<_  W )  /\  Q  .<_  X ) )  ->  -.  Q  .<_  W )
2 simpl1l 1009 . . . . . 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 30235 . . . . . 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 962 . . . . . 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 963 . . . . . 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 14485 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
104, 5, 6, 9syl3anc 1185 . . . . 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 740 . . . . . 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 30161 . . . . . 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 1010 . . . . . 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 30869 . . . . . 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 14496 . . . . 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 1187 . . . 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 449 . . . 4  |-  ( ( ( X  ./\  Y
)  .<_  W  /\  Q  .<_  W )  ->  Q  .<_  W )
2422, 23syl6bir 222 . . 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 171 . 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 735 . . . 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 32180 . . . 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 1193 . . 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 4225 . 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 294 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Latclat 14479   Atomscatm 30135   HLchlt 30222   LHypclh 30855
This theorem is referenced by:  dihjatc1  32183  dihmeetlem10N  32188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859
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