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Theorem llnmod2i2 30357
Description: Version of modular law pmod1i 30342 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 987 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  HL )
2 hllat 29858 . . . 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 989 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  e.  B )
5 simp2l 983 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  P  e.  A )
6 simp2r 984 . . . . 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 29861 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
111, 5, 6, 10syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( P  .\/  Q )  e.  B
)
12 simp12 988 . . . 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 14443 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  e.  B )
153, 11, 12, 14syl3anc 1184 . . 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 14451 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( ( P  .\/  Q )  ./\  X )  e.  B )  ->  ( Y  .\/  ( ( P 
.\/  Q )  ./\  X ) )  =  ( ( ( P  .\/  Q )  ./\  X )  .\/  Y ) )
173, 4, 15, 16syl3anc 1184 . 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 14442 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( Y  .\/  ( P  .\/  Q ) )  e.  B )
193, 4, 11, 18syl3anc 1184 . . . 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 14467 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Y  .\/  ( P 
.\/  Q ) )  e.  B )  -> 
( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P  .\/  Q ) )  ./\  X )
)
213, 12, 19, 20syl3anc 1184 . . 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 14451 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  Y  e.  B )  ->  (
( P  .\/  Q
)  .\/  Y )  =  ( Y  .\/  ( P  .\/  Q ) ) )
233, 11, 4, 22syl3anc 1184 . . . 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 6064 . . 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 959 . . . 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 30354 . . . 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 1206 . . 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 2454 . 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 14467 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X  ./\  ( P  .\/  Q ) )  =  ( ( P 
.\/  Q )  ./\  X ) )
313, 12, 11, 30syl3anc 1184 . . 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 6063 . 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 2455 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   lecple 13499   joincjn 14364   meetcmee 14365   Latclat 14437   Atomscatm 29758   HLchlt 29845
This theorem is referenced by:  dalawlem11  30375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-psubsp 29997  df-pmap 29998  df-padd 30290
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