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Theorem llnmod2i2 33870
Description: Version of modular law pmod1i 33855 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 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  HL )
2 hllat 33371 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  Lat )
4 simp13 1020 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  e.  B )
5 simp2l 1014 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  P  e.  A )
6 simp2r 1015 . . . . 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 33374 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
111, 5, 6, 10syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( P  .\/  Q )  e.  B
)
12 simp12 1019 . . . 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 15345 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  e.  B )
153, 11, 12, 14syl3anc 1219 . . 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 15352 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( ( P  .\/  Q )  ./\  X )  e.  B )  ->  ( Y  .\/  ( ( P 
.\/  Q )  ./\  X ) )  =  ( ( ( P  .\/  Q )  ./\  X )  .\/  Y ) )
173, 4, 15, 16syl3anc 1219 . 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 15344 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( Y  .\/  ( P  .\/  Q ) )  e.  B )
193, 4, 11, 18syl3anc 1219 . . . 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 15368 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Y  .\/  ( P 
.\/  Q ) )  e.  B )  -> 
( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P  .\/  Q ) )  ./\  X )
)
213, 12, 19, 20syl3anc 1219 . . 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 15352 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  Y  e.  B )  ->  (
( P  .\/  Q
)  .\/  Y )  =  ( Y  .\/  ( P  .\/  Q ) ) )
233, 11, 4, 22syl3anc 1219 . . . 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 6219 . . 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 990 . . . 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 33867 . . . 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 1241 . . 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 2505 . 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 15368 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X  ./\  ( P  .\/  Q ) )  =  ( ( P 
.\/  Q )  ./\  X ) )
313, 12, 11, 30syl3anc 1219 . . 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 6218 . 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 2506 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14296   lecple 14368   joincjn 15237   meetcmee 15238   Latclat 15338   Atomscatm 33271   HLchlt 33358
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 15239  df-plt 15251  df-lub 15267  df-glb 15268  df-join 15269  df-meet 15270  df-p0 15332  df-lat 15339  df-clat 15401  df-oposet 33184  df-ol 33186  df-oml 33187  df-covers 33274  df-ats 33275  df-atl 33306  df-cvlat 33330  df-hlat 33359  df-psubsp 33510  df-pmap 33511  df-padd 33803
This theorem is referenced by:  dalawlem11  33888
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