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Theorem atmod2i2 35059
Description: Version of modular law pmod2iN 35046 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 34561 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1017 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  K  e.  Lat )
3 simp21 1029 . . . . . 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 34487 . . . . . 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 1031 . . . . 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 15563 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  =  ( Y 
.\/  P ) )
112, 7, 8, 10syl3anc 1228 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  .\/  Y )  =  ( Y  .\/  P ) )
1211oveq1d 6310 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( ( P  .\/  Y )  ./\  X )  =  ( ( Y  .\/  P ) 
./\  X ) )
13 simp22 1030 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  X  e.  B )
144, 9latjcl 15555 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  e.  B )
152, 7, 8, 14syl3anc 1228 . . . 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 15579 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Y )  e.  B )  -> 
( X  ./\  ( P  .\/  Y ) )  =  ( ( P 
.\/  Y )  ./\  X ) )
182, 13, 15, 17syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( X  ./\  ( P  .\/  Y
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
19 simp1 996 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  K  e.  HL )
20 simp3 998 . . . 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 35056 . . . 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 1241 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  ./\  X
) )  =  ( ( Y  .\/  P
)  ./\  X )
)
2412, 18, 233eqtr4d 2518 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( X  ./\  ( P  .\/  Y
) )  =  ( Y  .\/  ( P 
./\  X ) ) )
254, 16latmcl 15556 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  ./\  X
)  e.  B )
262, 7, 13, 25syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  ./\ 
X )  e.  B
)
274, 9latjcom 15563 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  ./\  X )  e.  B )  -> 
( Y  .\/  ( P  ./\  X ) )  =  ( ( P 
./\  X )  .\/  Y ) )
282, 8, 26, 27syl3anc 1228 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  ./\  X
) )  =  ( ( P  ./\  X
)  .\/  Y )
)
294, 16latmcom 15579 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  ./\  X
)  =  ( X 
./\  P ) )
302, 7, 13, 29syl3anc 1228 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( P  ./\ 
X )  =  ( X  ./\  P )
)
3130oveq1d 6310 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  Y  .<_  X )  ->  ( ( P  ./\  X )  .\/  Y )  =  ( ( X  ./\  P )  .\/  Y ) )
3224, 28, 313eqtrrd 2513 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Latclat 15549   Atomscatm 34461   HLchlt 34548
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-psubsp 34700  df-pmap 34701  df-padd 34993
This theorem is referenced by:  llnexchb2lem  35065  dalawlem2  35069  dalawlem3  35070  dalawlem11  35078  dalawlem12  35079  cdleme15b  35472
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