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Theorem atmod1i1 30716
Description: Version of modular law pmod1i 30707 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 445 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  K  e.  HL )
2 simpr2 965 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
3 simpr1 964 . . . . 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 2438 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
8 eqid 2438 . . . . . 6  |-  ( + P `  K )  =  ( + P `  K )
94, 5, 6, 7, 8pmapjat2 30713 . . . . 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 1185 . . . 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 30149 . . . . 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 30715 . . . . 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 1237 . . . 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 657 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  ( P  .<_  Y  ->  (
( P  .\/  X
)  ./\  Y )  =  ( P  .\/  ( X  ./\  Y ) ) ) )
17163impia 1151 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( P  .\/  X )  ./\  Y )  =  ( P 
.\/  ( X  ./\  Y ) ) )
1817eqcomd 2443 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 set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   pmapcpmap 30356   + Pcpadd 30654
This theorem is referenced by:  atmod1i1m  30717  atmod2i1  30720  atmod3i1  30723  atmod4i1  30725  dalawlem6  30735  dalawlem11  30740  dalawlem12  30741  cdleme11g  31124  cdlemednpq  31158  cdleme20c  31170  cdleme22e  31203  cdleme22eALTN  31204  cdleme35c  31310
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-psubsp 30362  df-pmap 30363  df-padd 30655
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