Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atmod4i2 Structured version   Unicode version

Theorem atmod4i2 32848
Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-Mar-2013.)
Hypotheses
Ref Expression
atmod.b  |-  B  =  ( Base `  K
)
atmod.l  |-  .<_  =  ( le `  K )
atmod.j  |-  .\/  =  ( join `  K )
atmod.m  |-  ./\  =  ( meet `  K )
atmod.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
atmod4i2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( ( P  .\/  X ) 
./\  Y ) )

Proof of Theorem atmod4i2
StepHypRef Expression
1 hllat 32345 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1016 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 simp21 1028 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  A )
4 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
5 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 32271 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 17 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  B )
8 simp23 1030 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y  e.  B )
9 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
104, 9latmcl 15896 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  ./\  Y
)  e.  B )
112, 7, 8, 10syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( P  ./\ 
Y )  e.  B
)
12 simp22 1029 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
13 atmod.j . . . 4  |-  .\/  =  ( join `  K )
144, 13latjcom 15903 . . 3  |-  ( ( K  e.  Lat  /\  ( P  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( P  ./\  Y
)  .\/  X )  =  ( X  .\/  ( P  ./\  Y ) ) )
152, 11, 12, 14syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( X 
.\/  ( P  ./\  Y ) ) )
16 atmod.l . . 3  |-  .<_  =  ( le `  K )
174, 16, 13, 9, 5atmod1i2 32840 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
184, 13latjcom 15903 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  =  ( P 
.\/  X ) )
192, 12, 7, 18syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  P )  =  ( P  .\/  X ) )
2019oveq1d 6247 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( X  .\/  P )  ./\  Y )  =  ( ( P  .\/  X ) 
./\  Y ) )
2115, 17, 203eqtrd 2445 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( ( P  .\/  X ) 
./\  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1403    e. wcel 1840   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   meetcmee 15788   Latclat 15889   Atomscatm 32245   HLchlt 32332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-clat 15952  df-oposet 32158  df-ol 32160  df-oml 32161  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333  df-psubsp 32484  df-pmap 32485  df-padd 32777
This theorem is referenced by:  lhp2at0  33013  lhpelim  33018  cdleme2  33210  cdleme20yOLD  33285  cdleme35d  33435  cdlemeg46frv  33508  cdlemg2fv2  33583  cdlemg2m  33587  cdlemg10bALTN  33619  cdlemh2  33799  cdlemk9  33822  cdlemk9bN  33823
  Copyright terms: Public domain W3C validator