Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atmod2i2 Structured version   Unicode version

Theorem atmod2i2 33518
Description: Version of modular law pmod2iN 33505 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-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
atmod2i2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( ( X  ./\  P )  .\/  Y )  =  ( X 
./\  ( P  .\/  Y ) ) )

Proof of Theorem atmod2i2
StepHypRef Expression
1 hllat 33020 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  K  e.  Lat )
3 simp21 1021 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  P  e.  A )
4 atmod.b . . . . . . 7  |-  B  =  ( Base `  K
)
5 atmod.a . . . . . . 7  |-  A  =  ( Atoms `  K )
64, 5atbase 32946 . . . . . 6  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 16 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  P  e.  B )
8 simp23 1023 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  Y  e.  B )
9 atmod.j . . . . . 6  |-  .\/  =  ( join `  K )
104, 9latjcom 15241 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  =  ( Y 
.\/  P ) )
112, 7, 8, 10syl3anc 1218 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  .\/  Y )  =  ( Y  .\/  P ) )
1211oveq1d 6118 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( ( P  .\/  Y )  ./\  X )  =  ( ( Y  .\/  P ) 
./\  X ) )
13 simp22 1022 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  X  e.  B )
144, 9latjcl 15233 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  e.  B )
152, 7, 8, 14syl3anc 1218 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  .\/  Y )  e.  B
)
16 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
174, 16latmcom 15257 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Y )  e.  B )  -> 
( X  ./\  ( P  .\/  Y ) )  =  ( ( P 
.\/  Y )  ./\  X ) )
182, 13, 15, 17syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( X  ./\  ( P  .\/  Y
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
19 simp1 988 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  K  e.  HL )
20 simp3 990 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  Y  .<_  X )
21 atmod.l . . . . 5  |-  .<_  =  ( le `  K )
224, 21, 9, 16, 5atmod1i2 33515 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Y  e.  B  /\  X  e.  B
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  ./\  X
) )  =  ( ( Y  .\/  P
)  ./\  X )
)
2319, 3, 8, 13, 20, 22syl131anc 1231 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  ./\  X
) )  =  ( ( Y  .\/  P
)  ./\  X )
)
2412, 18, 233eqtr4d 2485 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( X  ./\  ( P  .\/  Y
) )  =  ( Y  .\/  ( P 
./\  X ) ) )
254, 16latmcl 15234 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  ./\  X
)  e.  B )
262, 7, 13, 25syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  ./\ 
X )  e.  B
)
274, 9latjcom 15241 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  ./\  X )  e.  B )  -> 
( Y  .\/  ( P  ./\  X ) )  =  ( ( P 
./\  X )  .\/  Y ) )
282, 8, 26, 27syl3anc 1218 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  ./\  X
) )  =  ( ( P  ./\  X
)  .\/  Y )
)
294, 16latmcom 15257 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  ./\  X
)  =  ( X 
./\  P ) )
302, 7, 13, 29syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  ./\ 
X )  =  ( X  ./\  P )
)
3130oveq1d 6118 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( ( P  ./\  X )  .\/  Y )  =  ( ( X  ./\  P )  .\/  Y ) )
3224, 28, 313eqtrrd 2480 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( ( X  ./\  P )  .\/  Y )  =  ( X 
./\  ( P  .\/  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   Basecbs 14186   lecple 14257   joincjn 15126   meetcmee 15127   Latclat 15227   Atomscatm 32920   HLchlt 33007
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-lat 15228  df-clat 15290  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-psubsp 33159  df-pmap 33160  df-padd 33452
This theorem is referenced by:  llnexchb2lem  33524  dalawlem2  33528  dalawlem3  33529  dalawlem11  33537  dalawlem12  33538  cdleme15b  33931
  Copyright terms: Public domain W3C validator