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Theorem dihmeetlem7N 36400
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 999 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  K  e.  HL )
3 hlatl 34450 . . . . . 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 1001 . . . . 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 2467 . . . . . 6  |-  ( 0.
`  K )  =  ( 0. `  K
)
11 dihmeetlem7.a . . . . . 6  |-  A  =  ( Atoms `  K )
127, 8, 9, 10, 11atnle 34407 . . . . 5  |-  ( ( K  e.  AtLat  /\  p  e.  A  /\  Y  e.  B )  ->  ( -.  p  .<_  Y  <->  ( p  ./\ 
Y )  =  ( 0. `  K ) ) )
134, 5, 6, 12syl3anc 1228 . . . 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 6310 . 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 34453 . . . . 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 1000 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  X  e.  B )
197, 9latmcl 15551 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
2017, 18, 6, 19syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( p  e.  A  /\  -.  p  .<_  Y ) )  ->  ( X  ./\ 
Y )  e.  B
)
217, 8, 9latmle2 15576 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  Y )
2217, 18, 6, 21syl3anc 1228 . . 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 34948 . . 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 1241 . 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 34451 . . . 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 34315 . . 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 2516 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4452   ` cfv 5593  (class class class)co 6294   Basecbs 14502   lecple 14574   joincjn 15443   meetcmee 15444   0.cp0 15536   Latclat 15544   OLcol 34264   Atomscatm 34353   AtLatcal 34354   HLchlt 34440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-lat 15545  df-clat 15607  df-oposet 34266  df-ol 34268  df-oml 34269  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-psubsp 34592  df-pmap 34593  df-padd 34885
This theorem is referenced by: (None)
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