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Theorem atmod4i1 33816
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
atmod4i1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( ( X  .\/  P ) 
./\  Y ) )

Proof of Theorem atmod4i1
StepHypRef Expression
1 hllat 33314 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  K  e.  Lat )
3 simp22 1022 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  X  e.  B )
4 simp23 1023 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  Y  e.  B )
5 atmod.b . . . . 5  |-  B  =  ( Base `  K
)
6 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
75, 6latmcl 15324 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
82, 3, 4, 7syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( X  ./\ 
Y )  e.  B
)
9 simp21 1021 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  P  e.  A )
10 atmod.a . . . . 5  |-  A  =  ( Atoms `  K )
115, 10atbase 33240 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
129, 11syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  P  e.  B )
13 atmod.j . . . 4  |-  .\/  =  ( join `  K )
145, 13latjcom 15331 . . 3  |-  ( ( K  e.  Lat  /\  ( X  ./\  Y )  e.  B  /\  P  e.  B )  ->  (
( X  ./\  Y
)  .\/  P )  =  ( P  .\/  ( X  ./\  Y ) ) )
152, 8, 12, 14syl3anc 1219 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( P 
.\/  ( X  ./\  Y ) ) )
16 atmod.l . . 3  |-  .<_  =  ( le `  K )
175, 16, 13, 6, 10atmod1i1 33807 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( ( P  .\/  X
)  ./\  Y )
)
185, 13latjcom 15331 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  X  e.  B )  ->  ( P  .\/  X
)  =  ( X 
.\/  P ) )
192, 12, 3, 18syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( P  .\/  X )  =  ( X  .\/  P ) )
2019oveq1d 6205 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( P  .\/  X )  ./\  Y )  =  ( ( X  .\/  P ) 
./\  Y ) )
2115, 17, 203eqtrd 2496 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( ( X  .\/  P ) 
./\  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   Latclat 15317   Atomscatm 33214   HLchlt 33301
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 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-psubsp 33453  df-pmap 33454  df-padd 33746
This theorem is referenced by:  dalawlem3  33823  dalawlem7  33827  dalawlem11  33831  cdleme9  34203  cdleme20aN  34259  cdleme22cN  34292  cdleme22d  34293  cdlemh1  34765  dia2dimlem1  35015  dia2dimlem2  35016  dia2dimlem3  35017
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