Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atmod2i1 Structured version   Unicode version

Theorem atmod2i1 34534
Description: Version of modular law pmod2iN 34522 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-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
atmod2i1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( X 
./\  ( Y  .\/  P ) ) )

Proof of Theorem atmod2i1
StepHypRef Expression
1 hllat 34037 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1012 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  Lat )
3 simp22 1025 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  X  e.  B )
4 simp23 1026 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  Y  e.  B )
5 atmod.b . . . . 5  |-  B  =  ( Base `  K
)
6 atmod.m . . . . 5  |-  ./\  =  ( meet `  K )
75, 6latmcom 15553 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
82, 3, 4, 7syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\ 
Y )  =  ( Y  ./\  X )
)
98oveq2d 6293 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( P  .\/  ( Y 
./\  X ) ) )
10 simp21 1024 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  e.  A )
11 atmod.a . . . . 5  |-  A  =  ( Atoms `  K )
125, 11atbase 33963 . . . 4  |-  ( P  e.  A  ->  P  e.  B )
1310, 12syl 16 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  e.  B )
145, 6latmcl 15530 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
152, 3, 4, 14syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\ 
Y )  e.  B
)
16 atmod.j . . . 4  |-  .\/  =  ( join `  K )
175, 16latjcom 15537 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( X  ./\  Y )  e.  B )  -> 
( P  .\/  ( X  ./\  Y ) )  =  ( ( X 
./\  Y )  .\/  P ) )
182, 13, 15, 17syl3anc 1223 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( ( X  ./\  Y
)  .\/  P )
)
195, 16latjcl 15529 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  e.  B )
202, 13, 4, 19syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  Y )  e.  B
)
215, 6latmcom 15553 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Y )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Y
)  ./\  X )  =  ( X  ./\  ( P  .\/  Y ) ) )
222, 20, 3, 21syl3anc 1223 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( ( P  .\/  Y )  ./\  X )  =  ( X 
./\  ( P  .\/  Y ) ) )
23 simp1 991 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  HL )
24 simp3 993 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  P  .<_  X )
25 atmod.l . . . . 5  |-  .<_  =  ( le `  K )
265, 25, 16, 6, 11atmod1i1 34530 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Y  e.  B  /\  X  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
2723, 10, 4, 3, 24, 26syl131anc 1236 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
285, 16latjcom 15537 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  P  e.  B )  ->  ( Y  .\/  P
)  =  ( P 
.\/  Y ) )
292, 4, 13, 28syl3anc 1223 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( Y  .\/  P )  =  ( P  .\/  Y ) )
3029oveq2d 6293 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\  ( Y  .\/  P
) )  =  ( X  ./\  ( P  .\/  Y ) ) )
3122, 27, 303eqtr4d 2513 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( X  ./\  ( Y  .\/  P ) ) )
329, 18, 313eqtr3d 2511 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( ( X  ./\  Y )  .\/  P )  =  ( X 
./\  ( Y  .\/  P ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   meetcmee 15423   Latclat 15523   Atomscatm 33937   HLchlt 34024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-psubsp 34176  df-pmap 34177  df-padd 34469
This theorem is referenced by:  lhpmod6i1  34712  trljat1  34839  trljat2  34840  cdlemc1  34864  cdlemc6  34869  cdleme16b  34952  cdleme20c  34984  cdleme20j  34991  cdleme22e  35017  cdleme22eALTN  35018  cdlemkid1  35595  dihmeetlem5  35982
  Copyright terms: Public domain W3C validator