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Theorem llnmod1i2 33516
Description: Version of modular law pmod1i 33504 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
llnmod1i2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  X  .<_  Y )  ->  ( X  .\/  ( ( P  .\/  Q )  ./\  Y )
)  =  ( ( X  .\/  ( P 
.\/  Q ) ) 
./\  Y ) )

Proof of Theorem llnmod1i2
StepHypRef Expression
1 simpl1 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
2 simpl2 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
3 simprl 755 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
4 simprr 756 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  A )
5 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
6 atmod.j . . . . . 6  |-  .\/  =  ( join `  K )
7 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
8 eqid 2443 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
9 eqid 2443 . . . . . 6  |-  ( +P `  K )  =  ( +P `  K )
105, 6, 7, 8, 9pmapjlln1 33511 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  P  e.  A  /\  Q  e.  A
) )  ->  (
( pmap `  K ) `  ( X  .\/  ( P  .\/  Q ) ) )  =  ( ( ( pmap `  K
) `  X )
( +P `  K ) ( (
pmap `  K ) `  ( P  .\/  Q
) ) ) )
111, 2, 3, 4, 10syl13anc 1220 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  (
( pmap `  K ) `  ( X  .\/  ( P  .\/  Q ) ) )  =  ( ( ( pmap `  K
) `  X )
( +P `  K ) ( (
pmap `  K ) `  ( P  .\/  Q
) ) ) )
12 hllat 33020 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
131, 12syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
145, 7atbase 32946 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
153, 14syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
165, 7atbase 32946 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
174, 16syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
185, 6latjcl 15233 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
1913, 15, 17, 18syl3anc 1218 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
20 simpl3 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  Y  e.  B )
21 atmod.l . . . . . 6  |-  .<_  =  ( le `  K )
22 atmod.m . . . . . 6  |-  ./\  =  ( meet `  K )
235, 21, 6, 22, 8, 9hlmod1i 33512 . . . . 5  |-  ( ( K  e.  HL  /\  ( X  e.  B  /\  ( P  .\/  Q
)  e.  B  /\  Y  e.  B )
)  ->  ( ( X  .<_  Y  /\  (
( pmap `  K ) `  ( X  .\/  ( P  .\/  Q ) ) )  =  ( ( ( pmap `  K
) `  X )
( +P `  K ) ( (
pmap `  K ) `  ( P  .\/  Q
) ) ) )  ->  ( ( X 
.\/  ( P  .\/  Q ) )  ./\  Y
)  =  ( X 
.\/  ( ( P 
.\/  Q )  ./\  Y ) ) ) )
241, 2, 19, 20, 23syl13anc 1220 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  (
( X  .<_  Y  /\  ( ( pmap `  K
) `  ( X  .\/  ( P  .\/  Q
) ) )  =  ( ( ( pmap `  K ) `  X
) ( +P `  K ) ( (
pmap `  K ) `  ( P  .\/  Q
) ) ) )  ->  ( ( X 
.\/  ( P  .\/  Q ) )  ./\  Y
)  =  ( X 
.\/  ( ( P 
.\/  Q )  ./\  Y ) ) ) )
2511, 24mpan2d 674 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  ( X  .<_  Y  ->  (
( X  .\/  ( P  .\/  Q ) ) 
./\  Y )  =  ( X  .\/  (
( P  .\/  Q
)  ./\  Y )
) ) )
26253impia 1184 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  X  .<_  Y )  ->  ( ( X  .\/  ( P  .\/  Q ) )  ./\  Y
)  =  ( X 
.\/  ( ( P 
.\/  Q )  ./\  Y ) ) )
2726eqcomd 2448 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
)  /\  X  .<_  Y )  ->  ( X  .\/  ( ( P  .\/  Q )  ./\  Y )
)  =  ( ( X  .\/  ( P 
.\/  Q ) ) 
./\  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4304   ` cfv 5430  (class class class)co 6103   Basecbs 14186   lecple 14257   joincjn 15126   meetcmee 15127   Latclat 15227   Atomscatm 32920   HLchlt 33007   pmapcpmap 33153   +Pcpadd 33451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-1st 6589  df-2nd 6590  df-poset 15128  df-plt 15140  df-lub 15156  df-glb 15157  df-join 15158  df-meet 15159  df-p0 15221  df-lat 15228  df-clat 15290  df-oposet 32833  df-ol 32835  df-oml 32836  df-covers 32923  df-ats 32924  df-atl 32955  df-cvlat 32979  df-hlat 33008  df-psubsp 33159  df-pmap 33160  df-padd 33452
This theorem is referenced by:  llnmod2i2  33519  dalawlem12  33538
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