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Theorem atmod1i1 35682
Description: Version of modular law pmod1i 35673 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 1003 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
) )  ->  X  e.  B )
3 simpr1 1002 . . . . 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 2457 . . . . . 6  |-  ( pmap `  K )  =  (
pmap `  K )
8 eqid 2457 . . . . . 6  |-  ( +P `  K )  =  ( +P `  K )
94, 5, 6, 7, 8pmapjat2 35679 . . . . 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 1228 . . . 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 35115 . . . . 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 35681 . . . . 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 1280 . . . 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 1193 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  Y )  ->  ( ( P  .\/  X )  ./\  Y )  =  ( P 
.\/  ( X  ./\  Y ) ) )
1817eqcomd 2465 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 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14643   lecple 14718   joincjn 15699   meetcmee 15700   Atomscatm 35089   HLchlt 35176   pmapcpmap 35322   +Pcpadd 35620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-psubsp 35328  df-pmap 35329  df-padd 35621
This theorem is referenced by:  atmod1i1m  35683  atmod2i1  35686  atmod3i1  35689  atmod4i1  35691  dalawlem6  35701  dalawlem11  35706  dalawlem12  35707  cdleme11g  36091  cdlemednpq  36125  cdleme20c  36138  cdleme22e  36171  cdleme22eALTN  36172  cdleme35c  36278
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