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Theorem atmod4i2 33869
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-Mar-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
atmod4i2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( ( P  .\/  X ) 
./\  Y ) )

Proof of Theorem atmod4i2
StepHypRef Expression
1 hllat 33366 . . . 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 simp21 1021 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  A )
4 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
5 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
64, 5atbase 33292 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
73, 6syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  B )
8 simp23 1023 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y  e.  B )
9 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
104, 9latmcl 15344 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  ./\  Y
)  e.  B )
112, 7, 8, 10syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( P  ./\ 
Y )  e.  B
)
12 simp22 1022 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
13 atmod.j . . . 4  |-  .\/  =  ( join `  K )
144, 13latjcom 15351 . . 3  |-  ( ( K  e.  Lat  /\  ( P  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( P  ./\  Y
)  .\/  X )  =  ( X  .\/  ( P  ./\  Y ) ) )
152, 11, 12, 14syl3anc 1219 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( X 
.\/  ( P  ./\  Y ) ) )
16 atmod.l . . 3  |-  .<_  =  ( le `  K )
174, 16, 13, 9, 5atmod1i2 33861 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
184, 13latjcom 15351 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  =  ( P 
.\/  X ) )
192, 12, 7, 18syl3anc 1219 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  P )  =  ( P  .\/  X ) )
2019oveq1d 6218 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( X  .\/  P )  ./\  Y )  =  ( ( P  .\/  X ) 
./\  Y ) )
2115, 17, 203eqtrd 2499 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( ( P  ./\  Y )  .\/  X )  =  ( ( P  .\/  X ) 
./\  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   meetcmee 15237   Latclat 15337   Atomscatm 33266   HLchlt 33353
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-psubsp 33505  df-pmap 33506  df-padd 33798
This theorem is referenced by:  lhp2at0  34034  lhpelim  34039  cdleme2  34230  cdleme20y  34304  cdleme35d  34454  cdlemeg46frv  34527  cdlemg2fv2  34602  cdlemg2m  34606  cdlemg10bALTN  34638  cdlemh2  34818  cdlemk9  34841  cdlemk9bN  34842
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