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Theorem atmod2i1 33503
Description: Version of modular law pmod2iN 33491 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 33006 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  Lat )
3 simp22 1022 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  X  e.  B )
4 simp23 1023 . . . 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 15244 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  =  ( Y 
./\  X ) )
82, 3, 4, 7syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\ 
Y )  =  ( Y  ./\  X )
)
98oveq2d 6106 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( P  .\/  ( Y 
./\  X ) ) )
10 simp21 1021 . . . 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 32932 . . . 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 15221 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
152, 3, 4, 14syl3anc 1218 . . 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 15228 . . 3  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  ( X  ./\  Y )  e.  B )  -> 
( P  .\/  ( X  ./\  Y ) )  =  ( ( X 
./\  Y )  .\/  P ) )
182, 13, 15, 17syl3anc 1218 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( X  ./\  Y
) )  =  ( ( X  ./\  Y
)  .\/  P )
)
195, 16latjcl 15220 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  .\/  Y
)  e.  B )
202, 13, 4, 19syl3anc 1218 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  Y )  e.  B
)
215, 6latmcom 15244 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Y )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Y
)  ./\  X )  =  ( X  ./\  ( P  .\/  Y ) ) )
222, 20, 3, 21syl3anc 1218 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( ( P  .\/  Y )  ./\  X )  =  ( X 
./\  ( P  .\/  Y ) ) )
23 simp1 988 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  K  e.  HL )
24 simp3 990 . . . 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 33499 . . . 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 1231 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( ( P  .\/  Y
)  ./\  X )
)
285, 16latjcom 15228 . . . . 5  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  P  e.  B )  ->  ( Y  .\/  P
)  =  ( P 
.\/  Y ) )
292, 4, 13, 28syl3anc 1218 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( Y  .\/  P )  =  ( P  .\/  Y ) )
3029oveq2d 6106 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( X  ./\  ( Y  .\/  P
) )  =  ( X  ./\  ( P  .\/  Y ) ) )
3122, 27, 303eqtr4d 2484 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  P  .<_  X )  ->  ( P  .\/  ( Y  ./\  X
) )  =  ( X  ./\  ( Y  .\/  P ) ) )
329, 18, 313eqtr3d 2482 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Basecbs 14173   lecple 14244   joincjn 15113   meetcmee 15114   Latclat 15214   Atomscatm 32906   HLchlt 32993
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-iin 4173  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-psubsp 33145  df-pmap 33146  df-padd 33438
This theorem is referenced by:  lhpmod6i1  33681  trljat1  33808  trljat2  33809  cdlemc1  33833  cdlemc6  33838  cdleme16b  33921  cdleme20c  33953  cdleme20j  33960  cdleme22e  33986  cdleme22eALTN  33987  cdlemkid1  34564  dihmeetlem5  34951
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