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Theorem llnmod2i2 30734
Description: Version of modular law pmod1i 30719 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 988 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  K  e.  HL )
2 hllat 30235 . . . 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 990 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  Y  e.  B )
5 simp2l 984 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  P  e.  A )
6 simp2r 985 . . . . 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 30238 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  B )
111, 5, 6, 10syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  Y  .<_  X )  ->  ( P  .\/  Q )  e.  B
)
12 simp12 989 . . . 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 14485 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
)  ./\  X )  e.  B )
153, 11, 12, 14syl3anc 1185 . . 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 14493 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( ( P  .\/  Q )  ./\  X )  e.  B )  ->  ( Y  .\/  ( ( P 
.\/  Q )  ./\  X ) )  =  ( ( ( P  .\/  Q )  ./\  X )  .\/  Y ) )
173, 4, 15, 16syl3anc 1185 . 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 14484 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( Y  .\/  ( P  .\/  Q ) )  e.  B )
193, 4, 11, 18syl3anc 1185 . . . 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 14509 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( Y  .\/  ( P 
.\/  Q ) )  e.  B )  -> 
( X  ./\  ( Y  .\/  ( P  .\/  Q ) ) )  =  ( ( Y  .\/  ( P  .\/  Q ) )  ./\  X )
)
213, 12, 19, 20syl3anc 1185 . . 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 14493 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  Y  e.  B )  ->  (
( P  .\/  Q
)  .\/  Y )  =  ( Y  .\/  ( P  .\/  Q ) ) )
233, 11, 4, 22syl3anc 1185 . . . 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 6100 . . 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 960 . . . 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 30731 . . . 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 1207 . . 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 2480 . 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 14509 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  ( P  .\/  Q )  e.  B )  -> 
( X  ./\  ( P  .\/  Q ) )  =  ( ( P 
.\/  Q )  ./\  X ) )
313, 12, 11, 30syl3anc 1185 . . 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 6099 . 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 2481 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Latclat 14479   Atomscatm 30135   HLchlt 30222
This theorem is referenced by:  dalawlem11  30752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-psubsp 30374  df-pmap 30375  df-padd 30667
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