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Theorem atmod3i2 30662
Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-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
atmod3i2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( Y  ./\  P
) )  =  ( Y  ./\  ( X  .\/  P ) ) )

Proof of Theorem atmod3i2
StepHypRef Expression
1 hllat 30161 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 simp23 992 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y  e.  B )
4 simp22 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
5 simp21 990 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  A )
6 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
7 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 30087 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
95, 8syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  B )
10 atmod.j . . . . 5  |-  .\/  =  ( join `  K )
116, 10latjcl 14479 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
122, 4, 9, 11syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  P )  e.  B
)
13 atmod.m . . . 4  |-  ./\  =  ( meet `  K )
146, 13latmcom 14504 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  .\/  P )  e.  B )  -> 
( Y  ./\  ( X  .\/  P ) )  =  ( ( X 
.\/  P )  ./\  Y ) )
152, 3, 12, 14syl3anc 1184 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Y  ./\  ( X  .\/  P
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
16 atmod.l . . 3  |-  .<_  =  ( le `  K )
176, 16, 10, 13, 7atmod1i2 30656 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
186, 13latmcom 14504 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  ./\  Y
)  =  ( Y 
./\  P ) )
192, 9, 3, 18syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( P  ./\ 
Y )  =  ( Y  ./\  P )
)
2019oveq2d 6097 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( X  .\/  ( Y 
./\  P ) ) )
2115, 17, 203eqtr2rd 2475 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( Y  ./\  P
) )  =  ( Y  ./\  ( X  .\/  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   Latclat 14474   Atomscatm 30061   HLchlt 30148
This theorem is referenced by:  dalawlem3  30670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-padd 30593
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