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Theorem atmod1i1 33501
Description: Version of modular law pmod1i 33492 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-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
atmod1i1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( ( P  .\/  X
)  ./\  Y )
)

Proof of Theorem atmod1i1
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  K  e.  HL )
2 simpr2 995 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
3 simpr1 994 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  P  e.  A )
4 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
5 atmod.j . . . . . 6  |-  .\/  =  ( join `  K )
6 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 eqid 2443 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
8 eqid 2443 . . . . . 6  |-  ( +P `  K )  =  ( +P `  K )
94, 5, 6, 7, 8pmapjat2 33498 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( pmap `  K
) `  ( P  .\/  X ) )  =  ( ( ( pmap `  K ) `  P
) ( +P `  K ) ( (
pmap `  K ) `  X ) ) )
101, 2, 3, 9syl3anc 1218 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( pmap `  K ) `  ( P  .\/  X
) )  =  ( ( ( pmap `  K
) `  P )
( +P `  K ) ( (
pmap `  K ) `  X ) ) )
114, 6atbase 32934 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
12 atmod.l . . . . . 6  |-  .<_  =  ( le `  K )
13 atmod.m . . . . . 6  |-  ./\  =  ( meet `  K )
144, 12, 5, 13, 7, 8hlmod1i 33500 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( P  .<_  Y  /\  ( ( pmap `  K
) `  ( P  .\/  X ) )  =  ( ( ( pmap `  K ) `  P
) ( +P `  K ) ( (
pmap `  K ) `  X ) ) )  ->  ( ( P 
.\/  X )  ./\  Y )  =  ( P 
.\/  ( X  ./\  Y ) ) ) )
1511, 14syl3anr1 1270 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( P  .<_  Y  /\  ( ( pmap `  K
) `  ( P  .\/  X ) )  =  ( ( ( pmap `  K ) `  P
) ( +P `  K ) ( (
pmap `  K ) `  X ) ) )  ->  ( ( P 
.\/  X )  ./\  Y )  =  ( P 
.\/  ( X  ./\  Y ) ) ) )
1610, 15mpan2d 674 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( P  .<_  Y  ->  (
( P  .\/  X
)  ./\  Y )  =  ( P  .\/  ( X  ./\  Y ) ) ) )
17163impia 1184 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( P  .\/  X )  ./\  Y )  =  ( P 
.\/  ( X  ./\  Y ) ) )
1817eqcomd 2448 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( ( P  .\/  X
)  ./\  Y )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   lecple 14245   joincjn 15114   meetcmee 15115   Atomscatm 32908   HLchlt 32995   pmapcpmap 33141   +Pcpadd 33439
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
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 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-lat 15216  df-clat 15278  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-psubsp 33147  df-pmap 33148  df-padd 33440
This theorem is referenced by:  atmod1i1m  33502  atmod2i1  33505  atmod3i1  33508  atmod4i1  33510  dalawlem6  33520  dalawlem11  33525  dalawlem12  33526  cdleme11g  33909  cdlemednpq  33943  cdleme20c  33955  cdleme22e  33988  cdleme22eALTN  33989  cdleme35c  34095
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