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Theorem llnmod1i2 30671
Description: Version of modular law pmod1i 30659 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 2296 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
9 eqid 2296 . . . . . 6  |-  ( + P `  K )  =  ( + P `  K )
105, 6, 7, 8, 9pmapjlln1 30666 . . . . 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 30175 . . . . . . 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 30101 . . . . . . 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 30101 . . . . . . 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 14172 . . . . . 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 30667 . . . . 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 2301 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 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   Atomscatm 30075   HLchlt 30162   pmapcpmap 30308   + Pcpadd 30606
This theorem is referenced by:  llnmod2i2  30674  dalawlem12  30693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607
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