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Theorem atmod3i2 30054
Description: Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-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
atmod3i2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( Y  ./\  P
) )  =  ( Y  ./\  ( X  .\/  P ) ) )

Proof of Theorem atmod3i2
StepHypRef Expression
1 hllat 29553 . . . 4  |-  ( K  e.  HL  ->  K  e.  Lat )
213ad2ant1 976 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  K  e.  Lat )
3 simp23 990 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  Y  e.  B )
4 simp22 989 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  X  e.  B )
5 simp21 988 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  A )
6 atmod.b . . . . . 6  |-  B  =  ( Base `  K
)
7 atmod.a . . . . . 6  |-  A  =  ( Atoms `  K )
86, 7atbase 29479 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
95, 8syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  P  e.  B )
10 atmod.j . . . . 5  |-  .\/  =  ( join `  K )
116, 10latjcl 14156 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  .\/  P
)  e.  B )
122, 4, 9, 11syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  P )  e.  B
)
13 atmod.m . . . 4  |-  ./\  =  ( meet `  K )
146, 13latmcom 14181 . . 3  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  ( X  .\/  P )  e.  B )  -> 
( Y  ./\  ( X  .\/  P ) )  =  ( ( X 
.\/  P )  ./\  Y ) )
152, 3, 12, 14syl3anc 1182 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( Y  ./\  ( X  .\/  P
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
16 atmod.l . . 3  |-  .<_  =  ( le `  K )
176, 16, 10, 13, 7atmod1i2 30048 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( ( X  .\/  P
)  ./\  Y )
)
186, 13latmcom 14181 . . . 4  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Y  e.  B )  ->  ( P  ./\  Y
)  =  ( Y 
./\  P ) )
192, 9, 3, 18syl3anc 1182 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( P  ./\ 
Y )  =  ( Y  ./\  P )
)
2019oveq2d 5874 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( P  ./\  Y
) )  =  ( X  .\/  ( Y 
./\  P ) ) )
2115, 17, 203eqtr2rd 2322 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  X  e.  B  /\  Y  e.  B
)  /\  X  .<_  Y )  ->  ( X  .\/  ( Y  ./\  P
) )  =  ( Y  ./\  ( X  .\/  P ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540
This theorem is referenced by:  dalawlem3  30062
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985
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