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Theorem llnmod1i2 30049
Description: Version of modular law pmod1i 30037 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 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  HL )
2 simpl2 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  X  e.  B )
3 simprl 732 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  P  e.  A )
4 simprr 733 . . . . 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 2283 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
9 eqid 2283 . . . . . 6  |-  ( + P `  K )  =  ( + P `  K )
105, 6, 7, 8, 9pmapjlln1 30044 . . . . 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 1184 . . . 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 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
131, 12syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  K  e.  Lat )
145, 7atbase 29479 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
153, 14syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  P  e.  B )
165, 7atbase 29479 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
174, 16syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  Q  e.  B )
185, 6latjcl 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
1913, 15, 17, 18syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( P  e.  A  /\  Q  e.  A
) )  ->  ( P  .\/  Q )  e.  B )
20 simpl3 960 . . . . 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 30045 . . . . 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 1184 . . . 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 655 . . 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 1148 . 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 2288 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   pmapcpmap 29686   + Pcpadd 29984
This theorem is referenced by:  llnmod2i2  30052  dalawlem12  30071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985
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