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Theorem llnmod1i2 33134
Description: Version of modular law pmod1i 33122 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 1008 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
2 simpl2 1009 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
3 simprl 762 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
4 simprr 764 . . . . 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 2429 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
9 eqid 2429 . . . . . 6  |-  ( +P `  K )  =  ( +P `  K )
105, 6, 7, 8, 9pmapjlln1 33129 . . . . 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 1266 . . . 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 32638 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
131, 12syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
145, 7atbase 32564 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
153, 14syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
165, 7atbase 32564 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
174, 16syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
185, 6latjcl 16248 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
1913, 15, 17, 18syl3anc 1264 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
20 simpl3 1010 . . . . 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 33130 . . . . 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 1266 . . . 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 678 . . 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 1202 . 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 2437 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15159   joincjn 16140   meetcmee 16141   Latclat 16242   Atomscatm 32538   HLchlt 32625   pmapcpmap 32771   +Pcpadd 33069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-lat 16243  df-clat 16305  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-psubsp 32777  df-pmap 32778  df-padd 33070
This theorem is referenced by:  llnmod2i2  33137  dalawlem12  33156
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