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Theorem atmod3i1 30588
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-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
atmod3i1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( X  ./\  ( P  .\/  Y ) ) )

Proof of Theorem atmod3i1
StepHypRef Expression
1 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  HL )
2 simp21 990 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  e.  A )
3 simp23 992 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  Y  e.  B )
4 simp22 991 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  X  e.  B )
5 simp3 959 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  .<_  X )
6 atmod.b . . . 4  |-  B  =  ( Base `  K
)
7 atmod.l . . . 4  |-  .<_  =  ( le `  K )
8 atmod.j . . . 4  |-  .\/  =  ( join `  K )
9 atmod.m . . . 4  |-  ./\  =  ( meet `  K )
10 atmod.a . . . 4  |-  A  =  ( Atoms `  K )
116, 7, 8, 9, 10atmod1i1 30581 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Y  e.  B  /\  X  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
121, 2, 3, 4, 5, 11syl131anc 1197 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
13 hllat 30088 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
14133ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  Lat )
156, 9latmcom 14496 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
1614, 4, 3, 15syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\ 
Y )  =  ( Y  ./\  X )
)
1716oveq2d 6089 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( P  .\/  ( Y 
./\  X ) ) )
186, 10atbase 30014 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
192, 18syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  e.  B )
206, 8latjcl 14471 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  e.  B )
2114, 19, 3, 20syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  Y )  e.  B
)
226, 9latmcom 14496 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Y )  e.  B )  -> 
( X  ./\  ( P  .\/  Y ) )  =  ( ( P 
.\/  Y )  ./\  X ) )
2314, 4, 21, 22syl3anc 1184 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\  ( P  .\/  Y
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
2412, 17, 233eqtr4d 2477 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( X  ./\  ( P  .\/  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29988   HLchlt 30075
This theorem is referenced by:  dalawlem2  30596  dalawlem3  30597  dalawlem6  30600  lhpmcvr3  30749  cdleme0cp  30938  cdleme0cq  30939  cdleme1  30951  cdleme4  30962  cdleme5  30964  cdleme8  30974  cdleme9  30977  cdleme10  30978  cdleme15b  30999  cdleme22e  31068  cdleme22eALTN  31069  cdleme23c  31075  cdleme35b  31174  cdleme35e  31177  cdleme42a  31195  trlcoabs2N  31446  cdlemi1  31542  cdlemk4  31558  dia2dimlem1  31789  dia2dimlem2  31790  cdlemn10  31931  dihglbcpreN  32025
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-psubsp 30227  df-pmap 30228  df-padd 30520
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