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Theorem atmod3i1 30126
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 955 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  HL )
2 simp21 988 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  e.  A )
3 simp23 990 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  Y  e.  B )
4 simp22 989 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  X  e.  B )
5 simp3 957 . . 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 30119 . . 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 1195 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
13 hllat 29626 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
14133ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  Lat )
156, 9latmcom 14183 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
1614, 4, 3, 15syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\ 
Y )  =  ( Y  ./\  X )
)
1716oveq2d 5876 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( P  .\/  ( Y 
./\  X ) ) )
186, 10atbase 29552 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
192, 18syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  e.  B )
206, 8latjcl 14158 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  e.  B )
2114, 19, 3, 20syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  Y )  e.  B
)
226, 9latmcom 14183 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Y )  e.  B )  -> 
( X  ./\  ( P  .\/  Y ) )  =  ( ( P 
.\/  Y )  ./\  X ) )
2314, 4, 21, 22syl3anc 1182 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\  ( P  .\/  Y
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
2412, 17, 233eqtr4d 2327 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 934    = wceq 1625    e. wcel 1686   class class class wbr 4025   ` cfv 5257  (class class class)co 5860   Basecbs 13150   lecple 13217   joincjn 14080   meetcmee 14081   Latclat 14153   Atomscatm 29526   HLchlt 29613
This theorem is referenced by:  dalawlem2  30134  dalawlem3  30135  dalawlem6  30138  lhpmcvr3  30287  cdleme0cp  30476  cdleme0cq  30477  cdleme1  30489  cdleme4  30500  cdleme5  30502  cdleme8  30512  cdleme9  30515  cdleme10  30516  cdleme15b  30537  cdleme22e  30606  cdleme22eALTN  30607  cdleme23c  30613  cdleme35b  30712  cdleme35e  30715  cdleme42a  30733  trlcoabs2N  30984  cdlemi1  31080  cdlemk4  31096  dia2dimlem1  31327  dia2dimlem2  31328  cdlemn10  31469  dihglbcpreN  31563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-psubsp 29765  df-pmap 29766  df-padd 30058
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