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Theorem atmod3i2 33832
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 33331 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 simp23 1023 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y  e.  B )
4 simp22 1022 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
5 simp21 1021 . . . . 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 33257 . . . . 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 15339 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
122, 4, 9, 11syl3anc 1219 . . 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 15363 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  .\/  P )  e.  B )  -> 
( Y  ./\  ( X  .\/  P ) )  =  ( ( X 
.\/  P )  ./\  Y ) )
152, 3, 12, 14syl3anc 1219 . 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 33826 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
186, 13latmcom 15363 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  ./\  Y
)  =  ( Y 
./\  P ) )
192, 9, 3, 18syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( P  ./\ 
Y )  =  ( Y  ./\  P )
)
2019oveq2d 6215 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( X  .\/  ( Y 
./\  P ) ) )
2115, 17, 203eqtr2rd 2502 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4399   ` cfv 5525  (class class class)co 6199   Basecbs 14291   lecple 14363   joincjn 15232   meetcmee 15233   Latclat 15333   Atomscatm 33231   HLchlt 33318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-poset 15234  df-plt 15246  df-lub 15262  df-glb 15263  df-join 15264  df-meet 15265  df-p0 15327  df-lat 15334  df-clat 15396  df-oposet 33144  df-ol 33146  df-oml 33147  df-covers 33234  df-ats 33235  df-atl 33266  df-cvlat 33290  df-hlat 33319  df-psubsp 33470  df-pmap 33471  df-padd 33763
This theorem is referenced by:  dalawlem3  33840
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