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Theorem dihmeetlem7N 34953
Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihmeetlem7.b  |-  B  =  ( Base `  K
)
dihmeetlem7.l  |-  .<_  =  ( le `  K )
dihmeetlem7.j  |-  .\/  =  ( join `  K )
dihmeetlem7.m  |-  ./\  =  ( meet `  K )
dihmeetlem7.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dihmeetlem7N  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( (
( X  ./\  Y
)  .\/  p )  ./\  Y )  =  ( X  ./\  Y )
)

Proof of Theorem dihmeetlem7N
StepHypRef Expression
1 simprr 756 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  -.  p  .<_  Y )
2 simpl1 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  K  e.  HL )
3 hlatl 33003 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  AtLat )
42, 3syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  K  e.  AtLat
)
5 simprl 755 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  p  e.  A )
6 simpl3 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  Y  e.  B )
7 dihmeetlem7.b . . . . . 6  |-  B  =  ( Base `  K
)
8 dihmeetlem7.l . . . . . 6  |-  .<_  =  ( le `  K )
9 dihmeetlem7.m . . . . . 6  |-  ./\  =  ( meet `  K )
10 eqid 2442 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 dihmeetlem7.a . . . . . 6  |-  A  =  ( Atoms `  K )
127, 8, 9, 10, 11atnle 32960 . . . . 5  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  Y  e.  B )  ->  ( -.  p  .<_  Y  <->  ( p  ./\ 
Y )  =  ( 0. `  K ) ) )
134, 5, 6, 12syl3anc 1218 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( -.  p  .<_  Y  <->  ( p  ./\ 
Y )  =  ( 0. `  K ) ) )
141, 13mpbid 210 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( p  ./\ 
Y )  =  ( 0. `  K ) )
1514oveq2d 6106 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( ( X  ./\  Y )  .\/  ( p  ./\  Y ) )  =  ( ( X  ./\  Y )  .\/  ( 0. `  K
) ) )
16 hllat 33006 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
172, 16syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  K  e.  Lat )
18 simpl2 992 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  X  e.  B )
197, 9latmcl 15221 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
2017, 18, 6, 19syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( X  ./\ 
Y )  e.  B
)
217, 8, 9latmle2 15246 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  Y )
2217, 18, 6, 21syl3anc 1218 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( X  ./\ 
Y )  .<_  Y )
23 dihmeetlem7.j . . . 4  |-  .\/  =  ( join `  K )
247, 8, 23, 9, 11atmod1i2 33501 . . 3  |-  ( ( K  e.  HL  /\  ( p  e.  A  /\  ( X  ./\  Y
)  e.  B  /\  Y  e.  B )  /\  ( X  ./\  Y
)  .<_  Y )  -> 
( ( X  ./\  Y )  .\/  ( p 
./\  Y ) )  =  ( ( ( X  ./\  Y )  .\/  p )  ./\  Y
) )
252, 5, 20, 6, 22, 24syl131anc 1231 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( ( X  ./\  Y )  .\/  ( p  ./\  Y ) )  =  ( ( ( X  ./\  Y
)  .\/  p )  ./\  Y ) )
26 hlol 33004 . . . 4  |-  ( K  e.  HL  ->  K  e.  OL )
272, 26syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  K  e.  OL )
287, 23, 10olj01 32868 . . 3  |-  ( ( K  e.  OL  /\  ( X  ./\  Y )  e.  B )  -> 
( ( X  ./\  Y )  .\/  ( 0.
`  K ) )  =  ( X  ./\  Y ) )
2927, 20, 28syl2anc 661 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( ( X  ./\  Y )  .\/  ( 0. `  K ) )  =  ( X 
./\  Y ) )
3015, 25, 293eqtr3d 2482 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( (
( X  ./\  Y
)  .\/  p )  ./\  Y )  =  ( X  ./\  Y )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Basecbs 14173   lecple 14244   joincjn 15113   meetcmee 15114   0.cp0 15206   Latclat 15214   OLcol 32817   Atomscatm 32906   AtLatcal 32907   HLchlt 32993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-psubsp 33145  df-pmap 33146  df-padd 33438
This theorem is referenced by: (None)
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