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Theorem llnmod1i2 34873
Description: Version of modular law pmod1i 34861 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 999 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
2 simpl2 1000 . . . . 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 2467 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
9 eqid 2467 . . . . . 6  |-  ( +P `  K )  =  ( +P `  K )
105, 6, 7, 8, 9pmapjlln1 34868 . . . . 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 1230 . . . 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 34377 . . . . . . 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 34303 . . . . . . 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 34303 . . . . . . 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 15541 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
1913, 15, 17, 18syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
20 simpl3 1001 . . . . 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 34869 . . . . 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 1230 . . . 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 1193 . 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 2475 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   Latclat 15535   Atomscatm 34277   HLchlt 34364   pmapcpmap 34510   +Pcpadd 34808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336  df-hlat 34365  df-psubsp 34516  df-pmap 34517  df-padd 34809
This theorem is referenced by:  llnmod2i2  34876  dalawlem12  34895
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