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Theorem llnmod2i2 32880
Description: Version of modular law pmod1i 32865 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join  P  .\/  Q). (Contributed by NM, 16-Sep-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
llnmod2i2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y
)  =  ( X 
./\  ( ( P 
.\/  Q )  .\/  Y ) ) )

Proof of Theorem llnmod2i2
StepHypRef Expression
1 simp11 1027 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  HL )
2 hllat 32381 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 17 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  Lat )
4 simp13 1029 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  e.  B )
5 simp2l 1023 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  P  e.  A )
6 simp2r 1024 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Q  e.  A )
7 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
8 atmod.j . . . . . 6  |-  .\/  =  ( join `  K )
9 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 32384 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
111, 5, 6, 10syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( P  .\/  Q )  e.  B
)
12 simp12 1028 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  X  e.  B )
13 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
147, 13latmcl 16006 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  e.  B )
153, 11, 12, 14syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( P  .\/  Q )  ./\  X )  e.  B )
167, 8latjcom 16013 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( ( P  .\/  Q )  ./\  X )  e.  B )  ->  ( Y  .\/  ( ( P 
.\/  Q )  ./\  X ) )  =  ( ( ( P  .\/  Q )  ./\  X )  .\/  Y ) )
173, 4, 15, 16syl3anc 1230 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( ( P  .\/  Q )  ./\  X )
)  =  ( ( ( P  .\/  Q
)  ./\  X )  .\/  Y ) )
187, 8latjcl 16005 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( Y  .\/  ( P  .\/  Q ) )  e.  B )
193, 4, 11, 18syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( P  .\/  Q
) )  e.  B
)
207, 13latmcom 16029 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Y  .\/  ( P 
.\/  Q ) )  e.  B )  -> 
( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P  .\/  Q ) )  ./\  X )
)
213, 12, 19, 20syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P 
.\/  Q ) ) 
./\  X ) )
227, 8latjcom 16013 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  Y  e.  B )  ->  (
( P  .\/  Q
)  .\/  Y )  =  ( Y  .\/  ( P  .\/  Q ) ) )
233, 11, 4, 22syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( P  .\/  Q )  .\/  Y )  =  ( Y 
.\/  ( P  .\/  Q ) ) )
2423oveq2d 6294 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( ( P  .\/  Q )  .\/  Y ) )  =  ( X 
./\  ( Y  .\/  ( P  .\/  Q ) ) ) )
25 simp3 999 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  .<_  X )
26 atmod.l . . . . 5  |-  .<_  =  ( le `  K )
277, 26, 8, 13, 9llnmod1i2 32877 . . . 4  |-  ( ( ( K  e.  HL  /\  Y  e.  B  /\  X  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( ( P  .\/  Q )  ./\  X )
)  =  ( ( Y  .\/  ( P 
.\/  Q ) ) 
./\  X ) )
281, 4, 12, 5, 6, 25, 27syl321anc 1252 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( Y  .\/  ( ( P  .\/  Q )  ./\  X )
)  =  ( ( Y  .\/  ( P 
.\/  Q ) ) 
./\  X ) )
2921, 24, 283eqtr4d 2453 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( ( P  .\/  Q )  .\/  Y ) )  =  ( Y 
.\/  ( ( P 
.\/  Q )  ./\  X ) ) )
307, 13latmcom 16029 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X  ./\  ( P  .\/  Q ) )  =  ( ( P 
.\/  Q )  ./\  X ) )
313, 12, 11, 30syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( X  ./\  ( P  .\/  Q
) )  =  ( ( P  .\/  Q
)  ./\  X )
)
3231oveq1d 6293 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y
)  =  ( ( ( P  .\/  Q
)  ./\  X )  .\/  Y ) )
3317, 29, 323eqtr4rd 2454 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( ( X  ./\  ( P  .\/  Q ) )  .\/  Y
)  =  ( X 
./\  ( ( P 
.\/  Q )  .\/  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Latclat 15999   Atomscatm 32281   HLchlt 32368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-psubsp 32520  df-pmap 32521  df-padd 32813
This theorem is referenced by:  dalawlem11  32898
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